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Generalized arboricity of graphs with large girth
Discrete Mathematics 342 (2019) 1343–1350
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Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Generalized arboricity of graphs with large girth✩ ∗
Tomasz Bartnicki a , Bartłomiej Bosek b , Sebastian Czerwiński a , , Michał Farnik c , Jarosław Grytczuk d , Zofia Miechowicz a a
Faculty of Mathematics, Computer Science, and Econometrics, University of Zielona Góra, 65-516 Zielona Góra, Poland Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, 30-348 Kraków, Poland c Faculty of Mathematics and Computer Science, Jagiellonian University, 30-348 Kraków, Poland d Faculty of Mathematics and Information Science, Warsaw University of Technology, 00-662 Warsaw, Poland b
article
info
Article history: Received 19 April 2018 Received in revised form 11 January 2019 Accepted 19 January 2019 Available online xxxx Keywords: Arboricity Acyclic chromatic index Bounded expansion Entropy compression
a b s t r a c t The arboricity of a graph G is the least number of colors needed to color the edges of G so that no cycle is monochromatic. We consider a higher order analog of this parameter, denoted by arbp (G), introduced recently by Nešetřil et al. (2014). It is defined as the least number of colors needed to color the edges of a graph so that each cycle C gets at least min{|C |, p + 1} colors. So, arb1 (G) is the usual arboricity of G, while arb2 (G) can be seen as a relaxed version of the acyclic chromatic index of G. By the results of Nešetřil et al. (2014) it follows that arbp (G) is bounded for classes of graphs with bounded expansion, provided that the girth of graphs G is sufficiently large (depending on p). Using more direct approach we obtain explicit upper bounds on arbp (G) for some classes of graphs. In particular, we prove that arbp (G) ≤ p + 1 for every planar graph of girth at least 2p+1 . A similar result holds for graphs with arbitrary fixed genus. We also demonstrate that arb2 (G) ≤ 5 for outerplanar graphs, which is best possible. By using entropy compression argument we prove that arbp (G) ≤ (∆ − 1)p + 1 for graphs of maximum degree ∆ and sufficiently large girth. © 2019 Elsevier B.V. All rights reserved.
1. Introduction The arboricity of a graph G, denoted arb(G), is the least number of colors needed to color the edges of G so that no cycle is monochromatic. By the well-known Nash–Williams’ theorem [17,18], arb(G) = ⌈f (G)⌉, where f (G) is the fractional arboricity of G, a quantity given by: f (G) = max H ⊆G
|E(H)| . |V (H)| − 1
In particular, arb(G) ≤ 3 for any simple planar graph G, since every such graph on n vertices has at most 3n − 6 edges. In this paper we consider a higher order analog of graph arboricity introduced recently by Nešetřil, Ossona de Mendez, and Zhu [15]. It is defined as follows. For a fixed positive integer p, a coloring of the edges of a graph G is called p-acyclic coloring if every cycle C in G contains at least min{|C |, p + 1} colors. The least number of colors needed for a p-acyclic ✩ Supported by The Polish National Science Centre Grant 2015/17/B/ST1/02660. ∗ Corresponding author. E-mail addresses: [email protected] (T. Bartnicki), [email protected] (B. Bosek), [email protected] (S. Czerwiński), [email protected] (M. Farnik), [email protected] (J. Grytczuk), [email protected] (Z. Miechowicz). https://doi.org/10.1016/j.disc.2019.01.018 0012-365X/© 2019 Elsevier B.V. All rights reserved.
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coloring of G is called the p-arboricity of a graph G, and denoted by arbp (G). Thus arb1 (G) = arb(G) is the standard arboricity, while arb2 (G) is a relaxed version of the acyclic chromatic index of G, denoted as a′ (G). This is a well studied graph invariant introduced independently by Fiamčik [5], and Alon, Sudakov, and Zaks [2], defined as the least number of colors in a proper edge coloring with no 2-colored cycles. In [7] Greenhill and Pikhurko studied a generalized acyclic chromatic index a′p (G) defined by a similar condition for cycles. In [15], Nešetřil, Ossona de Mendez, and Zhu proved a theorem characterizing classes of graphs with bounded p-arboricity in terms of expansion parameters. To state this result recall that the density of a multigraph M is a proportion of the number of edges to the number of vertices of M. Let us define fp (G) as the maximum density of a multigraph M whose subdivision by at most p − 1 vertices per edge is a subgraph of G. The main theorem of [15] asserts that arbp (G) and fp (G) are functionally equivalent. More precisely, every graph G satisfies (fp (G))1/p ≤ arbp (G) ≤ Pp (fp (G)), where Pp is a polynomial. This implies that for every class of graphs with bounded expansion, arbp (G) is bounded for graphs with sufficiently large girth (depending on p). Recall that a class of graphs has bounded expansion if for every fixed p ≥ 1, all simple topological minors of members of the class, obtained by subdividing each edge by at most p − 1 vertices, have bounded densities (see [14] for a variety of examples and alternative definitions). For instance, by this result we know that arbp (G) is bounded for planar graphs of sufficiently large girth, but no reasonable upper bound on arbp follows from the theorem. In this paper we demonstrate that p-arboricity of planar graphs attains the least possible value when girth is sufficiently large. Namely, we prove in Theorem 2 that arbp (G) ≤ p + 1 for every planar graph G of girth at least 2p+1 , for all p ≥ 2. We also prove a similar result for graphs of any fixed genus. Additionally for outerplanar graphs we prove that arb2 (G) ≤ 5, which is optimal. By using entropy compression argument we prove that arbp (G) ≤ (∆ − 1)p + 1 for every graph of maximum degree ∆ and sufficiently large girth (depending on p and ∆). The method of entropy compression is a refinement of the celebrated algorithmic version of the Lovász Local Lemma (see [1]), due to Moser and Tardos [13]. It has been successfully applied for many graph coloring problems, in particular for acyclic edge coloring of graphs (see [3,4,8,9]). 2. Basic observations and some exact values We start with a simple lower bound for arbp (G) of general graphs with girth at least p + 1 in terms of graph degeneracy. Recall that a graph G is d-degenerate if each subgraph of G contains vertex of degree at most d. Proposition 1. Every d-degenerate graph G of girth at least p + 1 satisfies arbp (G) >
1 pd. 2
Proof. Let p ≥ 1 be fixed, and let δ (G) denote the minimum degree of G. Assume that G has n vertices, m edges, and arbp (G) = k. Suppose that we are given some p-acyclic k-coloring of the edges of G. Let mi denote the number of edges in color i, for i = 1, 2, . . . , k. Since girth of G is at least p + 1, each cycle in G contains at least p + 1 colors. Therefore any p color classes form a forest in G. This implies that for any subset {i1 , . . . , ip } of p colors we have mi1 + · · · + mip < n. Summing up these inequalities over all p-subsets of colors we get
(
k−1
)
p−1
(m1 + · · · + mk ) <
( ) k
p
n,
which gives m < pk n. On the other hand nδ (G) ≤ 2m, which implies that k > δ (G) 2 . The last inequality holds for any subgraph of G. So, the proof is complete. □ p
One may suspect that the class of graphs with low arboricity has bounded arb2 . We show it is true for the classes of wheels and outerplanar graphs. The wheel Wn is a graph obtained from a cycle on n vertices (called the rim) by adding an extra vertex and n edges (called the spokes) and joining it with all vertices of the cycle. Clearly arb(Wn ) = 2 for each n ≥ 3. Proposition 2. If n is an even number then arb2 (Wn ) = 4 otherwise arb2 (Wn ) = 5 . Proof. Let E(Wn ) = R ∪ S be the set of edges of the wheel, where R is the set of edges belonging to the rim and S is the set of spokes. From Proposition 1 we have arb2 (Wn ) ≥ 4. Let n be even and let C = {1, 2, 3, 4} be a set of colors. First we precolor the edges of R with colors 1 and 2 alternately and the edges of S with colors 3 and 4 alternately. Such a coloring has one defect: the edges of the big cycle R have received only two colors. We can easily correct this defect by switching colors on two incident edges r ∈ R and s ∈ S. Suppose that n = 2k + 1. By a counting argument we show that four colors are not enough. Assume that W2k+1 has a 2-acyclic coloring and let a1 , a2 , a3 , a4 be the numbers of edges in each color. Since a1 + a2 + a3 + a4 = 4k + 2, there exists a pair ai ̸ = aj which satisfies ai + aj ≥ 2k + 2, but this contradicts the assumption that each pair of colors forms a forest which may have at most 2k + 1 edges. Finally we show that there exists a 2-acyclic coloring of an odd wheel with five colors. First we color two nonincident edges r ∈ R and s ∈ S with color 5, next we color the remaining edges of R with colors 1 and 2 alternately and the remaining edges of S with colors 3 and 4 alternately. It is easy to verify that such a coloring is 2-acyclic which completes the proof. □
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Table 1 Number of factorizations of K2n . 2n
4
6
8
10
12
14
f (2n) pf (2n)
1 1
1 1
6 1
396 1
526915620 5
1132835421602062347 23
Now we show that arb2 can be arbitrarily large in the class of graphs with arb1 (G) = 2. Proposition 3. Let n ≥ 2. Then for the complete bipartite graph K2,n we have
⌈√
arb2 (K2,n ) =
8(n − 1) + 1 + 3 2
⌉
.
Proof. Let {e1 , . . . , en , e′1 , . . . , e′n } be the set of edges of K2,n with each pair (ei , e′i ) having a common vertex of degree 2. Let C = {c1 , . . . , ck }, k ≥ 3, be the set of colors. Each cycle in K2,n has a form (ei , e′i , e′j , ej ) with i ̸ = j, hence each pair of edges ′ (ei , e′i ) must get different pair of colors from C or the common color of i ,) ei ) cannot be used on other edges. Also there (ka) pair(k(e −1 can be only one such monochromatic pair of edges, so we must have 2 ≥ 2 + 1 ≥ n. Solving this inequality with respect to the unknown integer k we get the lower bound
√ k≥
8(n − 1) + 1 + 3 2
.
Moreover, the smallest number of colors k which satisfies above inequality is also sufficient to color the edges of K2,n with 3 colors on each cycle. It gives the equality and completes the proof. □ Graphs K2,n are planar and arb(K2,n ) = 2 for n ≥ 2. Proposition 3 shows that arb2 cannot be generally bounded by any function of arb even if we restrict it only to planar graphs. It is similar for higher values of p. Let Ppn be a graph obtained from n edge-disjoint paths of length p having only their endpoints in common. It is obvious that for each n ≥ 2, arbp (Ppn+1 ) = p + 1. Indeed, in a p-acyclic coloring of Ppn+1 each path of length p + 1 can get all p + 1 available colors. However if we demand at least one more color than p + 1 on the cycles in Ppn+1 , then we need different ’’palette’’ of colors on each (p + 1)-path. Thus for high values of n we need ‘‘large’’ set of colors, so for each q > p we have lim arbq (Ppn+1 ) = ∞.
n→∞
3. Complete graphs Generally, the problem of finding the exact values of arb2 may be hard even for relatively simple classes of graphs. Observe that a 2-acyclic coloring of a complete graph is a proper one, since every triangle gets three different colors. Thus the edges in the same color create a matching in Kn . Obviously a proper coloring is not necessary 2-acyclic. From Proposition 1 we have the lower bound arb2 (Kn ) ≥ n. Surprisingly, the problem of determining the arb2 of complete graphs is equivalent to the famous conjecture of Kotzig [10] on a perfect 1-factorization. A 1-factor is a perfect matching, and a 1-factorization of a k-regular graph is an edge coloring with k colors. A perfect 1-factorization of a graph is a 1-factorization having the property that the union of every pair of 1-factors induces a Hamiltonian cycle. Conjecture 1 (Kotzig [10]). For any n ≥ 2, the edges of K2n can be decomposed into 2n − 1 perfect matchings such that the union of any two matchings forms a Hamiltonian cycle. So far it is known that the conjecture is true for two infinite classes of graphs K2p and Kp+1 , where p is an odd prime [20], and for some incidental values of 2n like
{16, 28, 36, 40, 50, 52, 126, 170, 244, 344, 730, 1332, 1370, 1850, 2198, 3126, 6860, 12168, 16808, 29792}. Table 1 suggests that Kotzig’s conjecture is hard. The function f counts the number of nonisomorphic 1-factorizations of complete graph and the function pf counts the number of nonisomorphic perfect 1-factorizations of the complete graph ([16], A000474, A120488). Proposition 4. For any n ≥ 2, the complete graph K2n has a perfect 1-factorization if and only if arb2 (K2n−1 ) = 2n − 1. Proof. Suppose that the graph K2n has a perfect 1-factorization with 2n − 1 perfect matchings. By coloring every perfect matching with different color and removing one vertex with all incident edges, we obtain a graph K2n−1 with a 2-acyclic coloring since every pair of colors induces a Hamiltonian path in this graph.
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Conversely, suppose that K2n−1 has a 2-acyclic coloring using 2n − 1 colors. Since this coloring is proper and the number of edges in K2n−1 is exactly (n − 1)(2n − 1), each color class must be a matching with n − 1 edges. Since there are no 2-colored cycles, every two color classes must induce a Hamiltonian path. Furthermore, each vertex ’’misses’’ an edge with exactly one color and each color is ’’missed’’ only once. By adding an extra vertex and joining it with all vertices of K2n−1 by the edges with missing color we create a properly colored graph K2n with a Hamiltonian cycle in every pair of colors. This coloring is the demanded perfect 1-factorization. □ If we remove another vertex from a colored K2n−1 we obtain a complete graph K2n−2 with a 2-acyclic coloring using 2n − 1 colors. In such a coloring only one color class can induce a perfect matching with n − 1 edges while all remaining classes contain n − 2 edges each. From a counting argument 2n − 1 is the optimal number of colors in a 2-acyclic coloring of K2n−2 . In this way we come to the following equivalent formulation of Kotzig’s conjecture. Conjecture 2 (Kotzig Equivalent). For any n ≥ 3, arb2 (K2n−1 ) = arb2 (K2n−2 ) = 2n − 1. 4. Outerplanar graphs An outerplanar graph is a graph that has a planar drawing for which all vertices belong to the outer face of the drawing. Alternatively, the outerplanar graphs are graphs without the minors K2,3 and K4 . Every outerplanar graph is 2-degenerate. Theorem 1. Let G be a maximal outerplanar graph with a fixed embedding on the plane. There exists an edge coloring of G by elements of Z5 such that every cycle contains at least three different colors. Moreover, if any cycle C contains exactly three colors, and these colors are consecutive elements of Z5 , say t − 1, t , t + 1, then no edge of C colored by t belongs to the outer face of G. In consequence, every outerplanar graph G satisfies arb2 (G) ≤ 5. Proof. We apply induction on the number of vertices. If n = 3 then we may color the three edges of G by three nonconsecutive colors, say 0, 1, 3. Let G be a maximal outerplanar graph with more than three vertices. Fix an embedding of G on the plane. Let v be a vertex of degree two in G, and let a and b be the two edges incident to v . Finally, let x denote the edge forming a triangle with edges a and b. Now, consider a graph G′ = G − v . By the induction assumption G′ has a coloring satisfying the assertion of the theorem. Let t be a color of the edge x in this coloring. We extend this coloring to G by coloring the edges a and b by colors t − 1 and t + 1 (in any of the two ways). We claim that the extended coloring satisfies the assertion of the theorem. Indeed, suppose on the contrary that there is a cycle C in G violating the statement at some point. Clearly, C must contain the vertex v and the two edges a and b. Notice that C is not the triangle formed of edges a, b, x, since the edge x colored by t does not belong to the outer face of G. So, we may assume that C has at least four vertices, and let C ′ be any cycle obtained from C by deleting edges a and b and adding edge x. Suppose first that C contains only the two colors t − 1 and t + 1. Then C ′ contains only the three colors t − 1, t , t + 1, but the color t is on the edge x belonging to the outer face of G′ , which contradicts the inductive assumption. Suppose now, that C contains exactly three colors that are consecutive elements of Z5 . Since it already has t − 1 and t + 1, the only possibility for the third color is t. But then, in the same way as before, the cycle C ′ contradicts the inductive assumption. The proof is complete. □ Notice that the proof of Theorem 1 actually gives an explicit algorithm producing a 2-acyclic coloring of any outerplanar graph with at most 5 colors. Our next result shows that this bound is best possible. Proposition 5. There exists an outerplanar graph G with arb2 (G) = 5. Proof. We construct an outerplanar graph which cannot be 2-acyclically colored with colors from the set {A, B, C , D}. Note that every maximal outerplanar graph G on n vertices has 2n − 3 edges. Suppose that G has a 2-acyclic 4-coloring of the edges, which can be represented as a partition: E(G) = A ∪ B ∪ C ∪ D. Let M = {a, b, c , d} denote the multiset of sizes of color classes in this partition, so, we have a + b + c + d = 2n − 3. We claim that for fixed n, the multiset M is uniquely determined and consists of numbers as equal as possible, that is: a = b = c and d = a ± 1 (up to renaming of colors). Indeed, it is not hard to see that otherwise, there would be two numbers that sum up to at least n. This would imply existence of a 2-colored cycle in G. In Table 2 we exhibit multisets M for some initial values of n. We start our construction from the triangle colored with colors A, B, C . In the next step we build 3 new triangles on each side of the initial triangle. It easy to color such obtained hexagon introducing the fourth color D on 3 added edges. Finally we repeat construction for the outer edges of hexagon. We add 6 new vertices and 12 new edges building 6 new triangles. Obtained graph G is maximal outerplanar hence it has 12 vertices and 21 edges.
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Table 2 Maximal outerplanar graphs.
|V |
|E |
Multisets M
3 6 9 10 12
3 9 15 17 21
{1, 1, 1, 0} {2, 2, 2, 3} {4, 4, 4, 3} {5, 4, 4, 4} {5, 5, 5, 6}
Suppose that the 2-acyclic coloring of the hexagon was extended to the whole graph G using only 4 colors. The multiset M of color class sizes is {5, 5, 5, 6} while it was {2, 2, 2, 3} on the hexagon (see Table 2). Consider the color X which is used 6 times on G. It was used on the new edges of G 3 or 4 times only, so 3 or 2 new vertices of G, respectively, are not incident with an edge colored with X . If we remove these vertices from G we obtain a maximal outerplanar graph on 9 or 10 vertices which is 2-acyclically colored with 4 colors, where color X is used 6 times. This contradicts the multiset patterns given in Table 2. □ Perhaps some modification of the algorithm from Theorem 1 could be useful to get an upper bound for p-arboricity of outerplanar graphs for any p ≥ 2. We verified by hand that if H is the hexagon mentioned in the proof of Proposition 5, then arb3 (H) = 7 and arb4 (H) = 9. This encourages us to make the following suppositions. Conjecture 3. If G is an outerplanar graph then arbp (G) ≤ 2p + 1 for any p ≥ 2. Conjecture 4. For any p ≥ 2 there exists an outerplanar graph G such that arbp (G) = 2p + 1. 5. Planar graphs We showed in Proposition 3 that arb2 is unbounded in the class of planar graphs. In this section we consider planar graphs without short cycles. Recall that the girth of a graph is the length of its shortest cycle. There is a lot of results on decompositions of planar graphs with sufficiently large girth. We apply the following one proved in [21]. Lemma 1 ([21]). Let G be a planar graph of girth at least 8. Then the set of edges of G can be decomposed into a forest and a matching. We will use this result in the proof of the following theorem. Theorem 2. Every planar graph G of girth at least 2p+1 satisfies arbp (G) = p + 1, for all p ≥ 2. Proof. The proof is by induction on p. First we prove the initial case p = 2. Let G be a planar graph with n vertices and girth at least 8. Let F and M denote the edges of a forest and a matching in decomposition of G guaranteed by Lemma 1, respectively. Let G/M denote the graph obtained from G by contracting the matching M. This graph is clearly planar of girth at least 4 (since at most half edges of any cycle in G may belong to M). By a standard application of Euler’s formula, every planar graph of order n and girth at least 4 has at most 2n − 4. Hence, by Nash–Williams’ theorem, the arboricity of G/M is at most 2. So, we may color the edges of forest F by red and blue so that no cycle of G/M is monochromatic. Then we color the edges of the matching M into green. We claim that no cycle C of G is bichromatic. Indeed, since C intersects the matching M, it contains green color. Moreover, the edges of C \ M form a cycle in G/M. Thus both colors, red and blue, are also present on C . This proves the theorem for p = 2. Now, let p > 2 be fixed, and assume that the assertion of the theorem holds up to p − 1. Let G be a planar graph of girth at least 2p+1 . As before, we start with decomposing G into a forest F and a matching M. Then we contract the edges of M thereby obtaining a graph G/M. Each cycle in G contains at most 2p edges of M, hence the girth of G/M is at least 2p . So, by inductive assumption, we may color the edges of G/M using p colors so that each cycle of G/M contains all p colors. Putting a new color on the edges of M gives a coloring of G with p + 1 colors on every cycle. The proof is complete. □ The girth 8 is optimal for decomposition from Lemma 1 (see [12]), though probably it is not optimal to achieve arb2 (G) = 3. On the other hand, the edges of the dodecahedron need four colors in a 2-acyclic coloring. Conjecture 5. Every planar graph G of girth at least 6 satisfies arb2 (G) = 3. A similar result can be proved for graphs of bounded genus. To see this we invoke the following general result from [12]. Lemma 2 ([12]). Let G be a graph with fractional arboricity satisfying f (G) ≤ 4/3. Then the set of edges of G can be decomposed into a forest and a matching.
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By general Euler formula we know that graphs of fixed genus γ and sufficiently large girth satisfy f (G) ≤ 4/3. Thus, for every fixed γ there exists gγ such that every graph of genus γ and girth at least gγ decomposes into a forest and a matching. Then arguing as in the two proofs above we get the following result. Theorem 3. Every graph of fixed genus γ and girth at least gγ · 2p−2 satisfies arbp (G) ≤ p + 1, for all p ≥ 2. The above results may suggest that perhaps large girth alone already guarantees boundedness of p-arboricity. However, this is not the case, which can be easily deduced from the following classical result of Erdős (see [1]). Lemma 3 (Erdős, 1959). For all k and l there exists a graph G with girth greater than k and χ (G) > l. It is not hard to see that graphs of large chromatic number cannot have small arboricity. To justify this connection, notice that if G is a d-degenerate graph, then χ (G) ≤ d + 1. Indeed, one may linearly order the vertices of G so that each vertex has at most d neighbors to the left, and then apply greedy coloring. On the other hand, by Proposition 1 we have d < 2 arb(G), which implies that χ (G) < 2 arb(G) + 1. Hence, by the result of Erdős it follows that there are graphs of arbitrarily large girth and arboricity (hence, p-arboricity too). In the next result we strengthen this observation. Theorem 4. In the class of graphs with arboricity 2 there exist graphs with arbitrarily large girth and arbitrarily large arb2 (G). Proof. Let G be a graph with large girth and large chromatic number, whose existence is guaranteed by Lemma 3. Note that arboricity of G is large as well. Take such graph G and G by one vertex to get a new graph G′ . This new √ subdivide each edge of ′ ′ graph has arboricity 2. We show that arb (G ) ≥ arb(G). Suppose that G has been 2-acyclically colored with k colors and 2 √ k < arb(G). Every subdivided edge in the corresponding graph G gets a pair of colors. But assigning to each pair of colors a new color we get the acyclic coloring of G with at most k2 < arb(G) colors. This contradiction completes the proof. □ 6. Graphs of bounded degree In this section we focus on bounded degree graphs. We denote by ∆(G) the maximum degree of a graph G. The result 2p of [15] gives a rather weak upper bound for arbp (G) of order roughly ∆2 . Much better bound follows from the results of [7] p+1
on generalized acyclic chromatic index, namely a′p (G) ≤ c ∆⌊ 2 ⌋ for every graph of maximum degree ∆, and some constant c. It is not surprising, however, that for graphs of large girth (depending on p and ∆) still better bound is true. As proved in [6], every graph of girth at least 3p∆ satisfies a′p (G) ≤ 6p∆. By using entropy compression method we further improve this bound for p-arboricity of bounded degree graphs with large girth. Before stating the result, let us briefly describe a basic idea, which is surprisingly simple and general. Suppose we are given a combinatorial structure G and we want to color its elements by some fixed set of K colors so as to avoid certain conflicts. To do that we pick a long sequence of colors C , and color the elements of G one by one (in some fixed order) using consecutive terms of C . Whenever a conflict appears we erase colors of some colored elements so as to get back to a proper partial coloring. Then we start again with coloring elements of G with further colors indicated by the sequence C . The process stops if either the sequence C has ended or if the whole structure has been successfully colored. Assume that the former case occurs for all sequences C (of some fixed length n). During execution of the procedure we register information on all erasures in a special table, which is called ‘‘log’’, so that when the process stops we are able to reconstruct the entire sequence C from the log and the final partial coloring c. This implies that the total number of color sequences C of length n is equal to the total number of possible outcomes (log, c). However it may happen that the latter number is bounded from above by some function which is strictly smaller than K n for large enough n. This contradiction means that the procedure had to stop before C has ended for at least one sequence C , giving a desired conflict-free coloring of the whole structure G. Theorem 5. For every pair of positive integers (p, ∆) there is an integer g(p, ∆) such that every graph G of maximum degree ∆ and girth g(G) ≥ g(p, ∆) satisfies arbp (G) ≤ (∆ − 1)p + 1. Proof. Let n be a large number. Suppose that G is a graph of maximum degree ∆ and girth g. We assume that G cannot be p-acyclic colored with K colors and we use a coloring–uncoloring procedure to encode a long string of colors C whose length is n. The procedure creates a log which can be used to recreate the original string. Thus K n is bounded by the number of possible logs. Before we start with the procedure we prepare the graph by ordering the set of edges and orienting all edges arbitrarily. n+ n The log consists of five elements: the first element is a binary sequence from the set {0, 1} g , the second one is a sequence n of sets of colors from {P ({1, 2, . . . , K })} g , the third one is a sequence from {1, 2, . . . , ∆ − 1}n , the fourth one is a sequence {1, 2, . . . , p}n , and the fifth one is a partial coloring of the graph. A completely colored cycle with at most p colors is called a p-cycle.
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We use the following procedure for i = 1, 2, . . . . , n: (1) color the first uncolored edge with the ith color in the sequence C , (2) append 1 to the first sequence of the log, (3) if there is a p-cycle, then for the p-cycle with the lexicographically smallest edges: (a) append 0 to the first sequence of the log, (b) encode the cycle (this will be explained below), (c) uncolor every edge of the cycle. After the final iteration we write down the obtained partial coloring of the graph in the log. Note that after finishing of each iteration there are no p-cycles. If the procedure uncolors a cycle, then the cycle must contain the edge that was colored in the current iteration. Indeed, in the first iteration there are no colored cycles, and when any p-cycles are created by coloring an edge, then in the same iteration one of them is uncolored together with that edge. Note also that coloring an edge might create several distinct p-cycles. However for any given set S of at most p colors only one cycle colored with colors from S can be created. Indeed, if there were two cycles colored with colors from S then their sum minus the edge would contain a p-cycle, which is a contradiction. We encode a cycle in the following way. We add the set of at most p colors from the cycle to the second sequence of the log. We encode the edges of the cycle starting with the edge that was colored as last. We go along the cycle in the direction indicated by the starting edge. At each vertex among the incident edges is the one which led us to the vertex, and at most ∆ − 1 other edges. We add to the third sequence of the log the number of the other edge in the cycle (with respect to the ordering of the edges) and then pass along this edge to the next vertex. After encoding the edges of the cycle we encode their colors in the fourth sequence of the log. We add the numbers of the colors of the edges in the set of at most p colors of the edges of the cycle. Observe that with the help of the log we can recreate the whole process of encoding together with the initial sequence of colors. Indeed, using the first and third sequence we can recreate which edges were colored or uncolored at each step. Afterwards we can reverse the procedure by starting from the final partial coloring and using the second and fourth sequence to color the edges of p-cycles. Now we will analyze the number of possible logs. First observe that each time we add 0 to the first sequence we uncolor at least g edges, hence there are at most gn zeros in the sequence and exactly n ones. By appending zeros at the end of sequences shorter than n + gn we obtain
(
n+
n g
)
n g
n
< (e(g + 1)) g
n
distinct sequences. For the second sequence there are obviously at most (K p ) g possibilities. For the third and fourth we obtain respectively (∆ − 1)n and pn . Finally the number of partial colorings is a number t independent of n. Thus we have n
K n ≤ (e(g + 1)) g K
pn g
(∆ − 1)n pn t ,
which implies p
K
1− g
1
1
≤ (e(g + 1)) g p(∆ − 1)t n .
By going to infinity with n we obtain 1
p
1+ g −p
K ≤ (e(g + 1)) g −p (p(∆ − 1))
,
which for g large enough is strictly less then p(∆ − 1) + 1. This completes the proof. □ A well known conjecture posed independently in [2] and [5] states that the acyclic chromatic index of a graph satisfies a′ (G) ≤ ∆ + 2. On the other hand, by Proposition 1 we have arb2 (G) ≥ ∆ + 1 for every ∆-regular graph. We therefore pose the following conjecture. Conjecture 6. For every fixed ∆ there is a number g = g(∆) such that every graph G of maximum degree ∆ and girth at least g satisfies arb2 (G) ≤ ∆ + 1. The conjecture is true for ∆ = 3 (see [11]). An upper bound from the above theorem is 5, but notice that the argument works also for the list version of the problem. 7. Final remarks The following general problem arises from our discussion. Let P be a fixed class of graphs with bounded expansion. Define Ap (P ) as the least integer k such that there is a number g = g(p, P ) for which every graph G ∈ P of girth at least g satisfies arbp (G) ≤ k. The least possible value for Ap (P ) is of course p + 1. We proved that this minimum is attained for planar graphs,
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and more generally, for graphs of any fixed genus. For which other bounded expansion classes this property holds? Clearly not for every one, since by Proposition 1 this cannot hold for regular graphs of degree at least three. However, the following conjecture seems plausible. Conjecture 7. Every proper minor-closed class of graphs M satisfies Ap (M) = p + 1. As proved by Thomassen [19], if M is such a class, then every member of M with sufficiently large girth is 2-degenerated, which harmonizes with the lower bound provided in Proposition 1. It would be nice to verify the conjecture at least for graphs of bounded treewidth. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
N. Alon, J.H. Spencer, The Probabilistic Method, John Wiley and Sons, Inc, 2000. N. Alon, B. Sudakov, A. Zaks, Acyclic edge colorings of graphs, J. Graph Theory 37 (2001) 157–167. V. Dujmović, G. Joret, J. Kozik, D.R. Wood, Nonrepetitive colouring via entropy compression, Combinatorica 36 (2016) 661–686. L. Esperet, A. Parreau, Acyclic edge colouring using entropy compression, European J. Combin. 34 (2013) 1019–1027. J. Fiamčik, Atsiklicheskij khromaticheskij klass grafa, Math. Slovaca 28 (1978) 139–145. S. Gerke, M. Raemy, Generalised acyclic edge colourings of graphs with large girth, Discrete Math. 307 (2007) 1668–1671. C. Greenhill, O. Pikhurko, Bounds on generalised acyclic chromatic numbers for bounded degree graphs, Graphs Combin. 21 (2005) 407–419. J. Grytczuk, J. Kozik, P. Micek, New approach to nonrepetitive sequences, Random Struct. Algorithms 42 (2013) 214–225. D.G. Harris, A. Srinivasan, Algorithmic and Enumerative Aspects of the Moser-Tardos Distribution, ACM Trans. Algorithms 13 (2017) Article No. 33. A. Kotzig, Hamilton graphs and hamilton circuits, theory of graphs and its applications, in: Proceedings of the Symposium of Smolenice 1963, Nakl.ČSAV, Praha, 1964, pp. 62–82. E. Mačajová, J. Mazák, An Algorithm for Optimal Acyclic Edge-Colouring of Cubic Graphs, in: FAW-AAIM, in: LNCS, 6681, 2011, pp. 136–146. M. Montassier, A. Raspaud P. Ossona de Mendez, X. Zhu, Decomposing a graph into forests, J. Combin. Theory Ser. B 102 (2012) 38–52. R.A. Moser, G. Tardos, A constructive proof of the general Lovasz Local Lemma, J. ACM 57 (2010) 15. J. Nešetřil, P. Ossona de Mendez, Sparsity, Springer, 2012. J. Nešetřil, P. Ossona de Mendez, X. Zhu, Colouring edges with many colours in cycles, J. Combin. Theory Ser. B 109 (2014) 102–119. N.J.E. Sloane, The OEIS, founded in 1964. C. St J.A. Nash-Williams, Edge-disjoint spanning trees of finite graphs, J. Lond. Math. Soc. 36 (1961) 445–450. C. St J.A. Nash-Williams, Decomposition of finite graphs into forests, J. Lond. Math. Soc. 39 (1964) 12. C. Thomassen, Paths, circuits and subdivisions, in: Selected Topics in Graph Theory, 3, Academic Press, New York, 1988, pp. 97–131. D.G. Wagner, On the perfect one-factorization conjecture, Discrete Math. 104 (1992) 211–215. Y. Wang, Q. Zhang, Decomposing a planar graph with girth at least 8 into a forest and a matching, Discrete Math. 311 (2011) 844–849.
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Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Generalized arboricity of graphs with large girth✩ ∗
Tomasz Bartnicki a , Bartłomiej Bosek b , Sebastian Czerwiński a , , Michał Farnik c , Jarosław Grytczuk d , Zofia Miechowicz a a
Faculty of Mathematics, Computer Science, and Econometrics, University of Zielona Góra, 65-516 Zielona Góra, Poland Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, 30-348 Kraków, Poland c Faculty of Mathematics and Computer Science, Jagiellonian University, 30-348 Kraków, Poland d Faculty of Mathematics and Information Science, Warsaw University of Technology, 00-662 Warsaw, Poland b
article
info
Article history: Received 19 April 2018 Received in revised form 11 January 2019 Accepted 19 January 2019 Available online xxxx Keywords: Arboricity Acyclic chromatic index Bounded expansion Entropy compression
a b s t r a c t The arboricity of a graph G is the least number of colors needed to color the edges of G so that no cycle is monochromatic. We consider a higher order analog of this parameter, denoted by arbp (G), introduced recently by Nešetřil et al. (2014). It is defined as the least number of colors needed to color the edges of a graph so that each cycle C gets at least min{|C |, p + 1} colors. So, arb1 (G) is the usual arboricity of G, while arb2 (G) can be seen as a relaxed version of the acyclic chromatic index of G. By the results of Nešetřil et al. (2014) it follows that arbp (G) is bounded for classes of graphs with bounded expansion, provided that the girth of graphs G is sufficiently large (depending on p). Using more direct approach we obtain explicit upper bounds on arbp (G) for some classes of graphs. In particular, we prove that arbp (G) ≤ p + 1 for every planar graph of girth at least 2p+1 . A similar result holds for graphs with arbitrary fixed genus. We also demonstrate that arb2 (G) ≤ 5 for outerplanar graphs, which is best possible. By using entropy compression argument we prove that arbp (G) ≤ (∆ − 1)p + 1 for graphs of maximum degree ∆ and sufficiently large girth. © 2019 Elsevier B.V. All rights reserved.
1. Introduction The arboricity of a graph G, denoted arb(G), is the least number of colors needed to color the edges of G so that no cycle is monochromatic. By the well-known Nash–Williams’ theorem [17,18], arb(G) = ⌈f (G)⌉, where f (G) is the fractional arboricity of G, a quantity given by: f (G) = max H ⊆G
|E(H)| . |V (H)| − 1
In particular, arb(G) ≤ 3 for any simple planar graph G, since every such graph on n vertices has at most 3n − 6 edges. In this paper we consider a higher order analog of graph arboricity introduced recently by Nešetřil, Ossona de Mendez, and Zhu [15]. It is defined as follows. For a fixed positive integer p, a coloring of the edges of a graph G is called p-acyclic coloring if every cycle C in G contains at least min{|C |, p + 1} colors. The least number of colors needed for a p-acyclic ✩ Supported by The Polish National Science Centre Grant 2015/17/B/ST1/02660. ∗ Corresponding author. E-mail addresses: [email protected] (T. Bartnicki), [email protected] (B. Bosek), [email protected] (S. Czerwiński), [email protected] (M. Farnik), [email protected] (J. Grytczuk), [email protected] (Z. Miechowicz). https://doi.org/10.1016/j.disc.2019.01.018 0012-365X/© 2019 Elsevier B.V. All rights reserved.
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coloring of G is called the p-arboricity of a graph G, and denoted by arbp (G). Thus arb1 (G) = arb(G) is the standard arboricity, while arb2 (G) is a relaxed version of the acyclic chromatic index of G, denoted as a′ (G). This is a well studied graph invariant introduced independently by Fiamčik [5], and Alon, Sudakov, and Zaks [2], defined as the least number of colors in a proper edge coloring with no 2-colored cycles. In [7] Greenhill and Pikhurko studied a generalized acyclic chromatic index a′p (G) defined by a similar condition for cycles. In [15], Nešetřil, Ossona de Mendez, and Zhu proved a theorem characterizing classes of graphs with bounded p-arboricity in terms of expansion parameters. To state this result recall that the density of a multigraph M is a proportion of the number of edges to the number of vertices of M. Let us define fp (G) as the maximum density of a multigraph M whose subdivision by at most p − 1 vertices per edge is a subgraph of G. The main theorem of [15] asserts that arbp (G) and fp (G) are functionally equivalent. More precisely, every graph G satisfies (fp (G))1/p ≤ arbp (G) ≤ Pp (fp (G)), where Pp is a polynomial. This implies that for every class of graphs with bounded expansion, arbp (G) is bounded for graphs with sufficiently large girth (depending on p). Recall that a class of graphs has bounded expansion if for every fixed p ≥ 1, all simple topological minors of members of the class, obtained by subdividing each edge by at most p − 1 vertices, have bounded densities (see [14] for a variety of examples and alternative definitions). For instance, by this result we know that arbp (G) is bounded for planar graphs of sufficiently large girth, but no reasonable upper bound on arbp follows from the theorem. In this paper we demonstrate that p-arboricity of planar graphs attains the least possible value when girth is sufficiently large. Namely, we prove in Theorem 2 that arbp (G) ≤ p + 1 for every planar graph G of girth at least 2p+1 , for all p ≥ 2. We also prove a similar result for graphs of any fixed genus. Additionally for outerplanar graphs we prove that arb2 (G) ≤ 5, which is optimal. By using entropy compression argument we prove that arbp (G) ≤ (∆ − 1)p + 1 for every graph of maximum degree ∆ and sufficiently large girth (depending on p and ∆). The method of entropy compression is a refinement of the celebrated algorithmic version of the Lovász Local Lemma (see [1]), due to Moser and Tardos [13]. It has been successfully applied for many graph coloring problems, in particular for acyclic edge coloring of graphs (see [3,4,8,9]). 2. Basic observations and some exact values We start with a simple lower bound for arbp (G) of general graphs with girth at least p + 1 in terms of graph degeneracy. Recall that a graph G is d-degenerate if each subgraph of G contains vertex of degree at most d. Proposition 1. Every d-degenerate graph G of girth at least p + 1 satisfies arbp (G) >
1 pd. 2
Proof. Let p ≥ 1 be fixed, and let δ (G) denote the minimum degree of G. Assume that G has n vertices, m edges, and arbp (G) = k. Suppose that we are given some p-acyclic k-coloring of the edges of G. Let mi denote the number of edges in color i, for i = 1, 2, . . . , k. Since girth of G is at least p + 1, each cycle in G contains at least p + 1 colors. Therefore any p color classes form a forest in G. This implies that for any subset {i1 , . . . , ip } of p colors we have mi1 + · · · + mip < n. Summing up these inequalities over all p-subsets of colors we get
(
k−1
)
p−1
(m1 + · · · + mk ) <
( ) k
p
n,
which gives m < pk n. On the other hand nδ (G) ≤ 2m, which implies that k > δ (G) 2 . The last inequality holds for any subgraph of G. So, the proof is complete. □ p
One may suspect that the class of graphs with low arboricity has bounded arb2 . We show it is true for the classes of wheels and outerplanar graphs. The wheel Wn is a graph obtained from a cycle on n vertices (called the rim) by adding an extra vertex and n edges (called the spokes) and joining it with all vertices of the cycle. Clearly arb(Wn ) = 2 for each n ≥ 3. Proposition 2. If n is an even number then arb2 (Wn ) = 4 otherwise arb2 (Wn ) = 5 . Proof. Let E(Wn ) = R ∪ S be the set of edges of the wheel, where R is the set of edges belonging to the rim and S is the set of spokes. From Proposition 1 we have arb2 (Wn ) ≥ 4. Let n be even and let C = {1, 2, 3, 4} be a set of colors. First we precolor the edges of R with colors 1 and 2 alternately and the edges of S with colors 3 and 4 alternately. Such a coloring has one defect: the edges of the big cycle R have received only two colors. We can easily correct this defect by switching colors on two incident edges r ∈ R and s ∈ S. Suppose that n = 2k + 1. By a counting argument we show that four colors are not enough. Assume that W2k+1 has a 2-acyclic coloring and let a1 , a2 , a3 , a4 be the numbers of edges in each color. Since a1 + a2 + a3 + a4 = 4k + 2, there exists a pair ai ̸ = aj which satisfies ai + aj ≥ 2k + 2, but this contradicts the assumption that each pair of colors forms a forest which may have at most 2k + 1 edges. Finally we show that there exists a 2-acyclic coloring of an odd wheel with five colors. First we color two nonincident edges r ∈ R and s ∈ S with color 5, next we color the remaining edges of R with colors 1 and 2 alternately and the remaining edges of S with colors 3 and 4 alternately. It is easy to verify that such a coloring is 2-acyclic which completes the proof. □
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Table 1 Number of factorizations of K2n . 2n
4
6
8
10
12
14
f (2n) pf (2n)
1 1
1 1
6 1
396 1
526915620 5
1132835421602062347 23
Now we show that arb2 can be arbitrarily large in the class of graphs with arb1 (G) = 2. Proposition 3. Let n ≥ 2. Then for the complete bipartite graph K2,n we have
⌈√
arb2 (K2,n ) =
8(n − 1) + 1 + 3 2
⌉
.
Proof. Let {e1 , . . . , en , e′1 , . . . , e′n } be the set of edges of K2,n with each pair (ei , e′i ) having a common vertex of degree 2. Let C = {c1 , . . . , ck }, k ≥ 3, be the set of colors. Each cycle in K2,n has a form (ei , e′i , e′j , ej ) with i ̸ = j, hence each pair of edges ′ (ei , e′i ) must get different pair of colors from C or the common color of i ,) ei ) cannot be used on other edges. Also there (ka) pair(k(e −1 can be only one such monochromatic pair of edges, so we must have 2 ≥ 2 + 1 ≥ n. Solving this inequality with respect to the unknown integer k we get the lower bound
√ k≥
8(n − 1) + 1 + 3 2
.
Moreover, the smallest number of colors k which satisfies above inequality is also sufficient to color the edges of K2,n with 3 colors on each cycle. It gives the equality and completes the proof. □ Graphs K2,n are planar and arb(K2,n ) = 2 for n ≥ 2. Proposition 3 shows that arb2 cannot be generally bounded by any function of arb even if we restrict it only to planar graphs. It is similar for higher values of p. Let Ppn be a graph obtained from n edge-disjoint paths of length p having only their endpoints in common. It is obvious that for each n ≥ 2, arbp (Ppn+1 ) = p + 1. Indeed, in a p-acyclic coloring of Ppn+1 each path of length p + 1 can get all p + 1 available colors. However if we demand at least one more color than p + 1 on the cycles in Ppn+1 , then we need different ’’palette’’ of colors on each (p + 1)-path. Thus for high values of n we need ‘‘large’’ set of colors, so for each q > p we have lim arbq (Ppn+1 ) = ∞.
n→∞
3. Complete graphs Generally, the problem of finding the exact values of arb2 may be hard even for relatively simple classes of graphs. Observe that a 2-acyclic coloring of a complete graph is a proper one, since every triangle gets three different colors. Thus the edges in the same color create a matching in Kn . Obviously a proper coloring is not necessary 2-acyclic. From Proposition 1 we have the lower bound arb2 (Kn ) ≥ n. Surprisingly, the problem of determining the arb2 of complete graphs is equivalent to the famous conjecture of Kotzig [10] on a perfect 1-factorization. A 1-factor is a perfect matching, and a 1-factorization of a k-regular graph is an edge coloring with k colors. A perfect 1-factorization of a graph is a 1-factorization having the property that the union of every pair of 1-factors induces a Hamiltonian cycle. Conjecture 1 (Kotzig [10]). For any n ≥ 2, the edges of K2n can be decomposed into 2n − 1 perfect matchings such that the union of any two matchings forms a Hamiltonian cycle. So far it is known that the conjecture is true for two infinite classes of graphs K2p and Kp+1 , where p is an odd prime [20], and for some incidental values of 2n like
{16, 28, 36, 40, 50, 52, 126, 170, 244, 344, 730, 1332, 1370, 1850, 2198, 3126, 6860, 12168, 16808, 29792}. Table 1 suggests that Kotzig’s conjecture is hard. The function f counts the number of nonisomorphic 1-factorizations of complete graph and the function pf counts the number of nonisomorphic perfect 1-factorizations of the complete graph ([16], A000474, A120488). Proposition 4. For any n ≥ 2, the complete graph K2n has a perfect 1-factorization if and only if arb2 (K2n−1 ) = 2n − 1. Proof. Suppose that the graph K2n has a perfect 1-factorization with 2n − 1 perfect matchings. By coloring every perfect matching with different color and removing one vertex with all incident edges, we obtain a graph K2n−1 with a 2-acyclic coloring since every pair of colors induces a Hamiltonian path in this graph.
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Conversely, suppose that K2n−1 has a 2-acyclic coloring using 2n − 1 colors. Since this coloring is proper and the number of edges in K2n−1 is exactly (n − 1)(2n − 1), each color class must be a matching with n − 1 edges. Since there are no 2-colored cycles, every two color classes must induce a Hamiltonian path. Furthermore, each vertex ’’misses’’ an edge with exactly one color and each color is ’’missed’’ only once. By adding an extra vertex and joining it with all vertices of K2n−1 by the edges with missing color we create a properly colored graph K2n with a Hamiltonian cycle in every pair of colors. This coloring is the demanded perfect 1-factorization. □ If we remove another vertex from a colored K2n−1 we obtain a complete graph K2n−2 with a 2-acyclic coloring using 2n − 1 colors. In such a coloring only one color class can induce a perfect matching with n − 1 edges while all remaining classes contain n − 2 edges each. From a counting argument 2n − 1 is the optimal number of colors in a 2-acyclic coloring of K2n−2 . In this way we come to the following equivalent formulation of Kotzig’s conjecture. Conjecture 2 (Kotzig Equivalent). For any n ≥ 3, arb2 (K2n−1 ) = arb2 (K2n−2 ) = 2n − 1. 4. Outerplanar graphs An outerplanar graph is a graph that has a planar drawing for which all vertices belong to the outer face of the drawing. Alternatively, the outerplanar graphs are graphs without the minors K2,3 and K4 . Every outerplanar graph is 2-degenerate. Theorem 1. Let G be a maximal outerplanar graph with a fixed embedding on the plane. There exists an edge coloring of G by elements of Z5 such that every cycle contains at least three different colors. Moreover, if any cycle C contains exactly three colors, and these colors are consecutive elements of Z5 , say t − 1, t , t + 1, then no edge of C colored by t belongs to the outer face of G. In consequence, every outerplanar graph G satisfies arb2 (G) ≤ 5. Proof. We apply induction on the number of vertices. If n = 3 then we may color the three edges of G by three nonconsecutive colors, say 0, 1, 3. Let G be a maximal outerplanar graph with more than three vertices. Fix an embedding of G on the plane. Let v be a vertex of degree two in G, and let a and b be the two edges incident to v . Finally, let x denote the edge forming a triangle with edges a and b. Now, consider a graph G′ = G − v . By the induction assumption G′ has a coloring satisfying the assertion of the theorem. Let t be a color of the edge x in this coloring. We extend this coloring to G by coloring the edges a and b by colors t − 1 and t + 1 (in any of the two ways). We claim that the extended coloring satisfies the assertion of the theorem. Indeed, suppose on the contrary that there is a cycle C in G violating the statement at some point. Clearly, C must contain the vertex v and the two edges a and b. Notice that C is not the triangle formed of edges a, b, x, since the edge x colored by t does not belong to the outer face of G. So, we may assume that C has at least four vertices, and let C ′ be any cycle obtained from C by deleting edges a and b and adding edge x. Suppose first that C contains only the two colors t − 1 and t + 1. Then C ′ contains only the three colors t − 1, t , t + 1, but the color t is on the edge x belonging to the outer face of G′ , which contradicts the inductive assumption. Suppose now, that C contains exactly three colors that are consecutive elements of Z5 . Since it already has t − 1 and t + 1, the only possibility for the third color is t. But then, in the same way as before, the cycle C ′ contradicts the inductive assumption. The proof is complete. □ Notice that the proof of Theorem 1 actually gives an explicit algorithm producing a 2-acyclic coloring of any outerplanar graph with at most 5 colors. Our next result shows that this bound is best possible. Proposition 5. There exists an outerplanar graph G with arb2 (G) = 5. Proof. We construct an outerplanar graph which cannot be 2-acyclically colored with colors from the set {A, B, C , D}. Note that every maximal outerplanar graph G on n vertices has 2n − 3 edges. Suppose that G has a 2-acyclic 4-coloring of the edges, which can be represented as a partition: E(G) = A ∪ B ∪ C ∪ D. Let M = {a, b, c , d} denote the multiset of sizes of color classes in this partition, so, we have a + b + c + d = 2n − 3. We claim that for fixed n, the multiset M is uniquely determined and consists of numbers as equal as possible, that is: a = b = c and d = a ± 1 (up to renaming of colors). Indeed, it is not hard to see that otherwise, there would be two numbers that sum up to at least n. This would imply existence of a 2-colored cycle in G. In Table 2 we exhibit multisets M for some initial values of n. We start our construction from the triangle colored with colors A, B, C . In the next step we build 3 new triangles on each side of the initial triangle. It easy to color such obtained hexagon introducing the fourth color D on 3 added edges. Finally we repeat construction for the outer edges of hexagon. We add 6 new vertices and 12 new edges building 6 new triangles. Obtained graph G is maximal outerplanar hence it has 12 vertices and 21 edges.
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Table 2 Maximal outerplanar graphs.
|V |
|E |
Multisets M
3 6 9 10 12
3 9 15 17 21
{1, 1, 1, 0} {2, 2, 2, 3} {4, 4, 4, 3} {5, 4, 4, 4} {5, 5, 5, 6}
Suppose that the 2-acyclic coloring of the hexagon was extended to the whole graph G using only 4 colors. The multiset M of color class sizes is {5, 5, 5, 6} while it was {2, 2, 2, 3} on the hexagon (see Table 2). Consider the color X which is used 6 times on G. It was used on the new edges of G 3 or 4 times only, so 3 or 2 new vertices of G, respectively, are not incident with an edge colored with X . If we remove these vertices from G we obtain a maximal outerplanar graph on 9 or 10 vertices which is 2-acyclically colored with 4 colors, where color X is used 6 times. This contradicts the multiset patterns given in Table 2. □ Perhaps some modification of the algorithm from Theorem 1 could be useful to get an upper bound for p-arboricity of outerplanar graphs for any p ≥ 2. We verified by hand that if H is the hexagon mentioned in the proof of Proposition 5, then arb3 (H) = 7 and arb4 (H) = 9. This encourages us to make the following suppositions. Conjecture 3. If G is an outerplanar graph then arbp (G) ≤ 2p + 1 for any p ≥ 2. Conjecture 4. For any p ≥ 2 there exists an outerplanar graph G such that arbp (G) = 2p + 1. 5. Planar graphs We showed in Proposition 3 that arb2 is unbounded in the class of planar graphs. In this section we consider planar graphs without short cycles. Recall that the girth of a graph is the length of its shortest cycle. There is a lot of results on decompositions of planar graphs with sufficiently large girth. We apply the following one proved in [21]. Lemma 1 ([21]). Let G be a planar graph of girth at least 8. Then the set of edges of G can be decomposed into a forest and a matching. We will use this result in the proof of the following theorem. Theorem 2. Every planar graph G of girth at least 2p+1 satisfies arbp (G) = p + 1, for all p ≥ 2. Proof. The proof is by induction on p. First we prove the initial case p = 2. Let G be a planar graph with n vertices and girth at least 8. Let F and M denote the edges of a forest and a matching in decomposition of G guaranteed by Lemma 1, respectively. Let G/M denote the graph obtained from G by contracting the matching M. This graph is clearly planar of girth at least 4 (since at most half edges of any cycle in G may belong to M). By a standard application of Euler’s formula, every planar graph of order n and girth at least 4 has at most 2n − 4. Hence, by Nash–Williams’ theorem, the arboricity of G/M is at most 2. So, we may color the edges of forest F by red and blue so that no cycle of G/M is monochromatic. Then we color the edges of the matching M into green. We claim that no cycle C of G is bichromatic. Indeed, since C intersects the matching M, it contains green color. Moreover, the edges of C \ M form a cycle in G/M. Thus both colors, red and blue, are also present on C . This proves the theorem for p = 2. Now, let p > 2 be fixed, and assume that the assertion of the theorem holds up to p − 1. Let G be a planar graph of girth at least 2p+1 . As before, we start with decomposing G into a forest F and a matching M. Then we contract the edges of M thereby obtaining a graph G/M. Each cycle in G contains at most 2p edges of M, hence the girth of G/M is at least 2p . So, by inductive assumption, we may color the edges of G/M using p colors so that each cycle of G/M contains all p colors. Putting a new color on the edges of M gives a coloring of G with p + 1 colors on every cycle. The proof is complete. □ The girth 8 is optimal for decomposition from Lemma 1 (see [12]), though probably it is not optimal to achieve arb2 (G) = 3. On the other hand, the edges of the dodecahedron need four colors in a 2-acyclic coloring. Conjecture 5. Every planar graph G of girth at least 6 satisfies arb2 (G) = 3. A similar result can be proved for graphs of bounded genus. To see this we invoke the following general result from [12]. Lemma 2 ([12]). Let G be a graph with fractional arboricity satisfying f (G) ≤ 4/3. Then the set of edges of G can be decomposed into a forest and a matching.
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By general Euler formula we know that graphs of fixed genus γ and sufficiently large girth satisfy f (G) ≤ 4/3. Thus, for every fixed γ there exists gγ such that every graph of genus γ and girth at least gγ decomposes into a forest and a matching. Then arguing as in the two proofs above we get the following result. Theorem 3. Every graph of fixed genus γ and girth at least gγ · 2p−2 satisfies arbp (G) ≤ p + 1, for all p ≥ 2. The above results may suggest that perhaps large girth alone already guarantees boundedness of p-arboricity. However, this is not the case, which can be easily deduced from the following classical result of Erdős (see [1]). Lemma 3 (Erdős, 1959). For all k and l there exists a graph G with girth greater than k and χ (G) > l. It is not hard to see that graphs of large chromatic number cannot have small arboricity. To justify this connection, notice that if G is a d-degenerate graph, then χ (G) ≤ d + 1. Indeed, one may linearly order the vertices of G so that each vertex has at most d neighbors to the left, and then apply greedy coloring. On the other hand, by Proposition 1 we have d < 2 arb(G), which implies that χ (G) < 2 arb(G) + 1. Hence, by the result of Erdős it follows that there are graphs of arbitrarily large girth and arboricity (hence, p-arboricity too). In the next result we strengthen this observation. Theorem 4. In the class of graphs with arboricity 2 there exist graphs with arbitrarily large girth and arbitrarily large arb2 (G). Proof. Let G be a graph with large girth and large chromatic number, whose existence is guaranteed by Lemma 3. Note that arboricity of G is large as well. Take such graph G and G by one vertex to get a new graph G′ . This new √ subdivide each edge of ′ ′ graph has arboricity 2. We show that arb (G ) ≥ arb(G). Suppose that G has been 2-acyclically colored with k colors and 2 √ k < arb(G). Every subdivided edge in the corresponding graph G gets a pair of colors. But assigning to each pair of colors a new color we get the acyclic coloring of G with at most k2 < arb(G) colors. This contradiction completes the proof. □ 6. Graphs of bounded degree In this section we focus on bounded degree graphs. We denote by ∆(G) the maximum degree of a graph G. The result 2p of [15] gives a rather weak upper bound for arbp (G) of order roughly ∆2 . Much better bound follows from the results of [7] p+1
on generalized acyclic chromatic index, namely a′p (G) ≤ c ∆⌊ 2 ⌋ for every graph of maximum degree ∆, and some constant c. It is not surprising, however, that for graphs of large girth (depending on p and ∆) still better bound is true. As proved in [6], every graph of girth at least 3p∆ satisfies a′p (G) ≤ 6p∆. By using entropy compression method we further improve this bound for p-arboricity of bounded degree graphs with large girth. Before stating the result, let us briefly describe a basic idea, which is surprisingly simple and general. Suppose we are given a combinatorial structure G and we want to color its elements by some fixed set of K colors so as to avoid certain conflicts. To do that we pick a long sequence of colors C , and color the elements of G one by one (in some fixed order) using consecutive terms of C . Whenever a conflict appears we erase colors of some colored elements so as to get back to a proper partial coloring. Then we start again with coloring elements of G with further colors indicated by the sequence C . The process stops if either the sequence C has ended or if the whole structure has been successfully colored. Assume that the former case occurs for all sequences C (of some fixed length n). During execution of the procedure we register information on all erasures in a special table, which is called ‘‘log’’, so that when the process stops we are able to reconstruct the entire sequence C from the log and the final partial coloring c. This implies that the total number of color sequences C of length n is equal to the total number of possible outcomes (log, c). However it may happen that the latter number is bounded from above by some function which is strictly smaller than K n for large enough n. This contradiction means that the procedure had to stop before C has ended for at least one sequence C , giving a desired conflict-free coloring of the whole structure G. Theorem 5. For every pair of positive integers (p, ∆) there is an integer g(p, ∆) such that every graph G of maximum degree ∆ and girth g(G) ≥ g(p, ∆) satisfies arbp (G) ≤ (∆ − 1)p + 1. Proof. Let n be a large number. Suppose that G is a graph of maximum degree ∆ and girth g. We assume that G cannot be p-acyclic colored with K colors and we use a coloring–uncoloring procedure to encode a long string of colors C whose length is n. The procedure creates a log which can be used to recreate the original string. Thus K n is bounded by the number of possible logs. Before we start with the procedure we prepare the graph by ordering the set of edges and orienting all edges arbitrarily. n+ n The log consists of five elements: the first element is a binary sequence from the set {0, 1} g , the second one is a sequence n of sets of colors from {P ({1, 2, . . . , K })} g , the third one is a sequence from {1, 2, . . . , ∆ − 1}n , the fourth one is a sequence {1, 2, . . . , p}n , and the fifth one is a partial coloring of the graph. A completely colored cycle with at most p colors is called a p-cycle.
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We use the following procedure for i = 1, 2, . . . . , n: (1) color the first uncolored edge with the ith color in the sequence C , (2) append 1 to the first sequence of the log, (3) if there is a p-cycle, then for the p-cycle with the lexicographically smallest edges: (a) append 0 to the first sequence of the log, (b) encode the cycle (this will be explained below), (c) uncolor every edge of the cycle. After the final iteration we write down the obtained partial coloring of the graph in the log. Note that after finishing of each iteration there are no p-cycles. If the procedure uncolors a cycle, then the cycle must contain the edge that was colored in the current iteration. Indeed, in the first iteration there are no colored cycles, and when any p-cycles are created by coloring an edge, then in the same iteration one of them is uncolored together with that edge. Note also that coloring an edge might create several distinct p-cycles. However for any given set S of at most p colors only one cycle colored with colors from S can be created. Indeed, if there were two cycles colored with colors from S then their sum minus the edge would contain a p-cycle, which is a contradiction. We encode a cycle in the following way. We add the set of at most p colors from the cycle to the second sequence of the log. We encode the edges of the cycle starting with the edge that was colored as last. We go along the cycle in the direction indicated by the starting edge. At each vertex among the incident edges is the one which led us to the vertex, and at most ∆ − 1 other edges. We add to the third sequence of the log the number of the other edge in the cycle (with respect to the ordering of the edges) and then pass along this edge to the next vertex. After encoding the edges of the cycle we encode their colors in the fourth sequence of the log. We add the numbers of the colors of the edges in the set of at most p colors of the edges of the cycle. Observe that with the help of the log we can recreate the whole process of encoding together with the initial sequence of colors. Indeed, using the first and third sequence we can recreate which edges were colored or uncolored at each step. Afterwards we can reverse the procedure by starting from the final partial coloring and using the second and fourth sequence to color the edges of p-cycles. Now we will analyze the number of possible logs. First observe that each time we add 0 to the first sequence we uncolor at least g edges, hence there are at most gn zeros in the sequence and exactly n ones. By appending zeros at the end of sequences shorter than n + gn we obtain
(
n+
n g
)
n g
n
< (e(g + 1)) g
n
distinct sequences. For the second sequence there are obviously at most (K p ) g possibilities. For the third and fourth we obtain respectively (∆ − 1)n and pn . Finally the number of partial colorings is a number t independent of n. Thus we have n
K n ≤ (e(g + 1)) g K
pn g
(∆ − 1)n pn t ,
which implies p
K
1− g
1
1
≤ (e(g + 1)) g p(∆ − 1)t n .
By going to infinity with n we obtain 1
p
1+ g −p
K ≤ (e(g + 1)) g −p (p(∆ − 1))
,
which for g large enough is strictly less then p(∆ − 1) + 1. This completes the proof. □ A well known conjecture posed independently in [2] and [5] states that the acyclic chromatic index of a graph satisfies a′ (G) ≤ ∆ + 2. On the other hand, by Proposition 1 we have arb2 (G) ≥ ∆ + 1 for every ∆-regular graph. We therefore pose the following conjecture. Conjecture 6. For every fixed ∆ there is a number g = g(∆) such that every graph G of maximum degree ∆ and girth at least g satisfies arb2 (G) ≤ ∆ + 1. The conjecture is true for ∆ = 3 (see [11]). An upper bound from the above theorem is 5, but notice that the argument works also for the list version of the problem. 7. Final remarks The following general problem arises from our discussion. Let P be a fixed class of graphs with bounded expansion. Define Ap (P ) as the least integer k such that there is a number g = g(p, P ) for which every graph G ∈ P of girth at least g satisfies arbp (G) ≤ k. The least possible value for Ap (P ) is of course p + 1. We proved that this minimum is attained for planar graphs,
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and more generally, for graphs of any fixed genus. For which other bounded expansion classes this property holds? Clearly not for every one, since by Proposition 1 this cannot hold for regular graphs of degree at least three. However, the following conjecture seems plausible. Conjecture 7. Every proper minor-closed class of graphs M satisfies Ap (M) = p + 1. As proved by Thomassen [19], if M is such a class, then every member of M with sufficiently large girth is 2-degenerated, which harmonizes with the lower bound provided in Proposition 1. It would be nice to verify the conjecture at least for graphs of bounded treewidth. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
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