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List strong edge-coloring of graphs with maximum degree 4
Discrete Mathematics 343 (2020) 111854
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Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
List strong edge-coloring of graphs with maximum degree 4 Baochen Zhang a , Yulin Chang a , Jie Hu b , Meijie Ma c , Donglei Yang a ,
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a
School of Mathematics, Shandong University, Jinan, Shandong, 250100, China Laboratoire de Recherche en Informatique, Université Paris-Sud, Orsay Cedex 91405, France c School of Management Science and Engineering, Shandong Technology and Business University, Yantai, 264005, China b
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Article history: Received 21 May 2018 Received in revised form 12 August 2019 Accepted 5 February 2020 Available online xxxx Keywords: List strong edge-coloring Combinatorial Nullstellensatz Hall’s Theorem
a b s t r a c t A strong edge-coloring of a graph G is an edge-coloring such that any two edges on a path of length three receive distinct colors. We denote the strong chromatic index by χs′ (G) which is the minimum number of colors that allow a strong edge-coloring of G. Erdős and Nešetřil conjectured in 1985 that the upper bound of χs′ (G) is 54 ∆2 when ∆ is even and 41 (5∆2 − 2∆ + 1) when ∆ is odd, where ∆ is the maximum degree of G. The conjecture is proved right when ∆ ≤ 3. The best known upper bound for ∆ = 4 is 22 (Cranston, 2006). In this paper we extend the result of Cranston to the list version, that is, we prove that when ∆ = 4, list strong chromatic index is at most 22. © 2020 Elsevier B.V. All rights reserved.
1. Introduction A strong edge-coloring is a proper edge-coloring such that no two edges on a path of length three have the same color. To be more precise, a strong k-edge-coloring of a graph G is a coloring φ : E(G) → [k] such that for any two edges e1 and e2 that are either adjacent to each other or adjacent to a common edge, φ (e1 ) ̸ = φ (e2 ). The strong chromatic index of G, denoted by χs′ (G), is the minimum positive integer k for which G has a strong k-edge-coloring. A graph G, together with a list assignment L = {L(e) : e ∈ E(G)}, is strongly L-edge-colorable if there exists a strong edge-coloring c for G such that c(e) ∈ L(e) for every edge e ∈ E(G). For a positive integer k, a graph G is strongly k-edgechoosable if G is strongly L-edge-colorable for every L with |L(e)| ≥ k for all e ∈ E(G). The strong choice number, denoted by χls′ (G), is the minimum positive integer k for which G is strongly k-edge-choosable. Given a graph G with the maximum degree ∆, Erdős and Nešetřil [6] conjectured that χs′ (G) is at most 45 ∆2 when ∆ is even and 41 (5∆2 − 2∆ + 1) when ∆ is odd. They also give a construction to show that if the conjecture is true, then the bound is tight. For graphs with ∆ = 3, the conjecture was proved by Andersen [2] and by Horák [8] independently. For ∆ = 4, the best known upper bound is 22, which is due to Cranston [4] . When ∆ is sufficiently large, Bonamy, Perrett and Postle [3] proved that χs′ (G) ≤ 1.835∆2 . As for k-degenerate graphs, Yu [12] proved that χs′ (G) ≤ (4k − 2)∆ − 2k2 + k + 1. More results can be found in [5,10,11]. In this paper, we mainly prove the following theorem which extends Cranston’s result [4] to the list version. Theorem 1.
Every graph G with maximum degree 4 is strongly 22-edge-choosable.
The rest of the paper is organized as follows. Section 2 introduces some basic notation and elementary tools. In Section 3, we explore some basic properties of the minimal counterexample to Theorem 1. We will complete the proof of Theorem 1 in Section 4. ∗ Corresponding author. E-mail address: [email protected] (D. Yang). https://doi.org/10.1016/j.disc.2020.111854 0012-365X/© 2020 Elsevier B.V. All rights reserved.
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Fig. 1. The neighborhood of e.
2. Preliminaries and notation Throughout this paper, we assume that G is connected and is allowed to contain loops and multiple edges. The girth of G, denoted by g(G), is the length of the shortest cycle in G. Let l-cycle be a cycle of length l. For any distinct vertices u and v , the distance dist(u, v ) is the length of the shortest path connecting u and v . For a vertex v and a cycle C , let dist(v, C ) := min{dist(v, u) | u ∈ V (C )}. In particular, for any two distinct edges e1 and e2 , let dist(e1 , e2 ) := min{distG (u, v ) | u ∈ e1 , v ∈ e2 }. The neighborhood NG (e) of an edge e is the set of edges e′ such that dist(e, e′ ) ≤ 1. If the graph G is clear from the context, then we will omit the subscript. A partial list edge coloring of G is an edge coloring c for a proper subgraph H of G with c(e) ∈ L(e) for each edge e ∈ E(H). Given a partial list edge coloring c and an edge e, a color in L(e) is available for e if the color is not used on any edges in N(e), and denote by L′ (e) the set of available colors for e. Let N c (e) be the set of edges in N(e) that have been colored in the coloring c. In particular, |N(e)| ≤ 24 for an edge e as shown in Fig. 1. In all figures of this paper, the black vertices have no other neighbors than those appeared while the white vertices may have other neighbors. One of the main tools we use is the Combinatorial Nullstellensatz due to Alon [1].
∑n
Lemma 2.1 ([1]). Let F be an arbitrary field, and let P = P(x1 , . . . , xn ) be a polynomial in F[x1 , . . . , xn ]. If deg(P) = i=1 ki , k k where each ki is a non-negative integer, and the coefficient of x11 · · · xnn in P is non-zero, then for any subsets S1 , . . . , Sn of F with |Si | > ki for each i ∈ [n], there exist s1 ∈ S1 , . . . , sn ∈ Sn such that P(s1 , . . . , sn ) ̸ = 0. Another tool is the well-known Hall’s Theorem. Lemma 2.2 ([7]). Let A1 , . . . , An be n subsets of a set U. Then there is a set {a1 , . . . , an } of distinct elements in U with ai ∈ Ai for each i ∈ [n] if and only if for any integer k with 1 ≤ k ≤ n and every subcollection {Ai1 , . . . , Aik } of subsets, we have |Ai1 ∪ · · · ∪ Aik | ≥ k. Suppose that we have a partial list coloring c of G with an edge set T uncolored. Let L′ (e) be the set of available colors for each e ∈ T . If c cannot ⋃ be extended to E(G) by coloring ⋃ each edge e ∈ T , then by Lemma 2.2, there exists a subset S ⊆ T such that |S | > | e∈S L′ (e)|. Define disc(S) = |S | − | e∈S L′ (e)| to be the discrepancy of S. The following lemma is a generalization of Lemma 1 in [4]. Lemma 2.3. Let T be the set of uncolored edges in a partially colored graph and S be a subset of T with maximum discrepancy. Then any list strong edge-coloring of S can be extended to the entire T . Proof. Assume there exists a list strong edge-coloring c of S that cannot be extended to T . Hence there exists ⋃ a subset ′ S ′ ⊆ T⋃ \S with positive discrepancy (under the coloring c). We claim that disc(S ∪ S ′ ) > disc(S). Let R = e∈S ∪S ′ L (e), ⋃ R1 = e∈S L′ (e), k = disc(S) and R2 = e∈S ′ L′′ (e), where L′′ (e) denotes the set of available colors for each e ∈ S ′ under the coloring c. Then |S | = k + |R1 | and |S ′ | ≥ 1 + |R2 |. Since S and S ′ are disjoint, we get
|S ∪ S ′ | = |S | + |S ′ | ≥ k + 1 + |R1 | + |R2 | > k + |R|. The latter inequality follows from the fact that |R| = |R1 ∪ R2 | ≤ |R1 |+|R2 |. Hence disc(S ∪ S ′ ) = |S ∪ S ′ |−|R| > k = disc(S). This contradicts the maximality of disc(S). Thus, any coloring of S can be extended to the entire T . □ In the end of this section, we prove a useful lemma for the main proof in Section 4. Lemma 2.4. Let G be a graph with maximum degree 4, and v be an arbitrary vertex in G, then G − v is strongly 21-edge-choosable. If C is a cycle of length at least 3 in G, then G − E(C ) is strongly 21-edge-choosable. Proof. We consider the first case G −v . Define a partition V (G) = V1 ∪ V2 . . .∪ Vs by letting Vi = {u | dist(u, v ) = i} for each i ∈ [s], where s is the largest distance from v to all the other vertices. Therefore this yields an ordering σ = (e1 , e2 , . . . , em )
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Fig. 2. The situation in the proof of Lemma 2.4.
of all edges of G − v such that dist(v, ei ) ≤ dist(v, ej ) for any integers 1 ≤ i < j ≤ m. We now color the edges from em back to e1 one by one in the ordering σ . Once we meet an edge e = xy with dist(x, v ) = l ≤ dist(y, v ). Let u be a neighbor of x which is in Vl−1 (see Fig. 2). Then none of the edges incident with u has been colored. Since |N c (e)| ≤ 20, we can always find a color available for e. We proceed the proof of the second part when the edges of the cycle C are left uncolored. Similarly, we define a partition V (G) = V1 ∪ V2 . . . ∪ Vs by letting Vi = {u | dist(u, C ) = i} for each i ∈ [s], which yields an ordering σ = (e1 , e2 , . . . , em′ ) of all edges of G − E(C ) such that dist(v, ei ) ≤ dist(v, ej ) for any integers 1 ≤ i < j ≤ m′ . We now color the edges from em′ back to e1 one by one in the ordering σ . By similar arguments as in the case G − v , we can greedily color every edge that is not incident with C . Let e be an edge incident to C . If |C | ≥ 4, then |N(e) ∩ E(C )| ≥ 4; so again |L′ (e)| ≥ 21−(24−4) = 1. If |C | = 3, then |N(e)| ≤ 23. The three uncolored edges of C imply that |L′ (e)| ≥ 21−(23−3) = 1. Thus we can easily obtain a desired strong edge-coloring for G. This completes the proof of Lemma 2.4. □ 3. Basic properties Let G be an edge-minimal counterexample to Theorem 1, which means there exists a list assignment L with |L(e)| = 22 for each e ∈ E(G) such that G is not strongly L-edge-colorable while any proper subgraph of G is strongly L-edge-colorable. In this section, we show that G is a simple 4-regular graph and g(G) ≥ 6. Claim 3.1.
G is 4-regular.
Proof. Suppose G is not 4-regular. Let v be a vertex of G with d(v ) ≤ 3. By Lemma 2.4, we have a partial list coloring c with all the incident edges e1 , . . . , et of v uncolored, where t ≤ 3. Hence, for each i ∈ [t ], |N c (ei )| ≤ 18, which means |L′ (ei )| ≥ 22 − 18 = 4. One can easily extend c to a strong edge-coloring for G, a contradiction. □ Claim 3.2.
G is simple.
Proof. Suppose G is not simple. If G has a loop e1 incident with a vertex v , then let e2 , e3 be the edges incident with v which are not loops. By Lemma 2.4, we have a partial list coloring c such that the three incident edges e1 , e2 , e3 of v are uncolored. Similarly we have |L′ (e1 )| ≥ 22 − 6 = 18, |L′ (e2 )| ≥ 22 − 15 = 7, |L′ (e3 )| ≥ 7, and one can easily extend c to the entire graph. Next we consider the other case when G has multiple edges. Let v be a vertex in a 2-cycle containing two edges e1 , e2 and e3 , e4 be the incident edges of v that are not in the 2-cycle. By Lemma 2.4, we have a partial list coloring c for G − v . Similarly we have |L′ (ei )| ≥ 22 − 14 = 8 for i = 1, 2 and |L′ (ej )| ≥ 22 − 17 = 5 for j = 3, 4. We can easily extend c to a strong edge-coloring for G. □ Claim 3.3.
G has no 3-cycle.
Proof. Suppose G has a 3-cycle C . By Lemma 2.4, we can color all edges except the edges of C . It follows that |L′ (e)| ≥ 22 − 18 = 4 for each e ∈ E(C ) and we can easily obtain a strong edge-coloring for G. □ Claim 3.4.
G has no 4-cycle.
Proof. Suppose G has a 4-cycle C with all edges labeled in Fig. 3(a). We call all the edges labeled ai and bi pendant edges. We say two pendant edges form an adjacent pair if they share an endpoint which is not in C . The only possibility is that a1 or b1 shares an endpoint with a3 or b3 (or similarly a2 or b2 shares an endpoint with a4 or b4 ). In particular, we call {a1 , b1 , a3 , b3 } (or {a2 , b2 , a4 , b4 }) a pack. If there are at least two adjacent pairs, then by Lemma 2.4, we have a partial list coloring c for G − E(C ). It follows that |L′ (ci )| ≥ 22 − 18 = 4 for each i ∈ [4]. So we can easily extend c to the entire graph. In this situation, there is at least one pack in which there is no adjacent pair, and each edge connecting two pendant edges in a pack is called a diagonal edge. Hence there are at most 4 diagonal edges for a pack (see Fig. 3(b)). We now claim that there are at most 3 diagonal edges in a pack. Indeed, if there exist 4 diagonal edges for a pack, say {a1 , b1 , a3 , b3 }, then the neighborhood of each diagonal edge has size at most 21. By Lemma 2.4, we can color all edges except the four diagonal
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Fig. 3. 4-cycle.
edges. Thus each uncolored diagonal edge e satisfies |L′ (e)| ≥ 22 − (21 − 3) = 4, and it follows that one can easily obtain a desired strong edge-coloring for G, a contradiction. By the minimality of G, G − V (C ) admits a strongly edge-coloring c. Then we consider the following two cases. (1) There is exactly one adjacent pair, say a2 and a4 . Then there exists an pair of edges in the pack {a1 , b1 , a3 , b3 }, say a1 and a3 , such that there is no diagonal edge connecting a1 , a3 . ′ Observe that |L′ (ci )| ≥ 11 for each i ∈ [4], |L′ (ai )| ≥ 7 and i = 1, 3. In addition, |L′ (b2 )| ≥ 8, |L′ (b4 )| ≥ ⋃|4L (bi )′ | ≥ 7 for ′ ′ ′ 8, |L (a2 )| ≥ 11 and |L (a4 )| ≥ 11. If there is a color x ∈ i=1 (L (ai ) ∪ L (bi )), but x ∈ / L(cj ) for some j ∈ [4] (suppose that color x ∈ L′ (a1 ) \ L′ (c1 )), then we can give a1 color x and color the remaining pendant edges by Lemma 2.4. After this, we have |L′ (ci )| ≥ 3 for i = 2, 3, 4 and |L′ (c1 )| ≥ 4, and we obtain a desired strong edge-coloring for G by coloring the four edges in the order c4 , c3 , c2 , c1 . Similarly, if |L′ (ci )| > 11 for some i ∈ [4], then we can also obtain a desired coloring by the same strategy as above. ⋃4 ′ ′ ′ ′ Thus it remains to consider the case when i=1 (L (ai ) ∪ L (bi )) ⊆ L (cj ) and |L (cj )| = 11 for each j ∈ [4]. Since ′ ′ ′ ′ |L (a1 )| ≥ 7 and |L (a3 )| ≥ 7, there are at least 3 colors in L (a1 ) ∩ L (a3 ). We assign a1 and a3 with one color y ∈ L′ (a1 ) ∩ L′ (a3 ) and color the remaining pendant edges by Lemma 2.4. After this, it holds that |L′ (ci )| ≥ 4 for each i ∈ [4]. Thus we have a desired strong edge-coloring for G. (2) There is no adjacent pair. Since |L′ (ai )| ≥ 7, |L′ (bi )| ≥ 7 and |L′ (ci )| ≥ 10 for each i ∈ [4]. By Lemma 2.2 applied to the 12 uncolored edges, if we cannot assign a distinct color to each edge, then there is a subset S of the 12 uncolored edges with maximum positive discrepancy. We may assume that S contains an edge ci for some i ∈ [4], otherwise we can color S by Lemma 2.4, then extend the partial coloring to the remaining uncolored edges by Lemma 2.3. Since disc(S) > 0 and |L′ (ci )| ≥ 10, it follows that |S | is 11 or 12. If |S | = 12, then the two packs {a1 , b1 , a3 , b3 } and {a2 , b2 , a4 , b4 } are all in S. Recall that each pack has a pair of nonadjacent edges with no diagonal edge between them, suppose that the two pairs are (a1 , a3 ) and (a2 , b4 ). Since |L′ (a1 )| ≥ 7, |L′ (a3 )| ≥ 7, and |L′ (a1 ) ∪ L′ (a3 )| ≤ |S | − disc(S) ≤ 11, we have |L′ (a1 ) ∩ L′ (a3 )| ≥ 3. Similarly, we have |L′ (a2 ) ∩ L′ (b4 )| ≥ 3. Then we choose two distinct colors x, y with x ∈ L′ (a1 ) ∩ L′ (a3 ), y ∈ L′ (a2 ) ∩ L′ (b4 ) and assign a1 , a3 with color x while assign a2 and b4 with color y. After coloring the remaining pendant edges by Lemma 2.4, we have |L′ (ci )| ≥ 4 for each i ∈ [4] and we are done. If |S | = 11, then there is an uncolored edge e such that e ∈ / S. We may further assume that e is a pendant edge, otherwise we can color the 12 edges by the same argument as the case |S | = 12. Suppose that e = a1 . Since the pack {a2 , b2 , a4 , b4 } has a pair of nonadjacent edges with no diagonal edge between them, say (a2 , a4 ). Then again we can assign a2 , a4 with the same color and color the remaining pendant edges in S by Lemma 2.4. In this situation, a1 is uncolored and the resulting available color set L′′ (ci ) for each ci satisfies |L′′ (ci )| ≥ 10 − 7 + 1 = 4. Then one can easily obtain a partial coloring for S and extend it to a final coloring for the 12 edges by Lemma 2.3. This completes the proof of Claim 3.4. □ Claim 3.5.
G has no 5-cycle.
Proof. Suppose G has a 5-cycle C shown in Fig. 4(a). We also refer to the edges labeled by ai and bi as pendant edges. We claim that at least one of a4 and b4 is not in the neighborhood of b2 , otherwise we have a 4-cycle. Similarly, we assume that there is no edge between the following pairs: (b1 , b3 ), (b2 , b5 ) and (b5 , b3 ). We color all edges except five edges on C by Lemma 2.4 and then erase the colors of b2 , b3 , b4 , b5 . The resulting coloring implies that |L′ (ci )| ≥ 5 for i = 1, 2, 4, 5,
B. Zhang, Y. Chang, J. Hu et al. / Discrete Mathematics 343 (2020) 111854
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Fig. 4. 5-cycle.
Fig. 5. Proof of Theorem 1: Final analysis.
|L′ (c3 )| ≥ 6, |L′ (b2 )| ≥ 3, |L′ (b5 )| ≥ 3 and |L′ (b3 )| ≥ 4, |L′ (b4 )| ≥ 4. We relabel the edges as shown in Fig. 4(b). Hence we will finish the coloring if there exist si ∈ L′ (xi ) for each i ∈ [9] such that P(s1 , . . . , sn ) ̸ = 0 where ∏ ∏ ∏ P(x1 , x2 , . . . , x9 ) = (x1 − xi ) (x2 − xj ) (x3 − xk ) 2≤i≤7
3≤j≤8
4≤k≤9
(x1 − x9 )(x4 − x5 )(x4 − x7 )(x4 − x8 )(x4 − x9 ) (x5 − x6 )(x5 − x8 )(x5 − x9 )(x6 − x7 )(x7 − x8 ) (x8 − x9 ). We use MATLAB to calculate the coefficients of specific monomials. The codes are listed in the last section. By MATLAB, we obtain a coefficient cP (x31 x42 x53 x44 x45 x26 x37 x28 x29 ) = −1 ̸ = 0. Since deg(P) = 29, Lemma 2.1 implies that there exist si ∈ L′ (xi ) for each i ∈ [9] such that P(s1 , . . . , s9 ) ̸ = 0. Then we obtain a strong edge-coloring for G by coloring x1 , . . . , x9 with s1 , . . . , s9 respectively, a contradiction. □ 4. Final analyis In this section, we complete the proof of Theorem 1. Recall that G is a minimal counterexample to Theorem 1 and by Claim 3.1–3.5, G is a simple 4-regular graph with g(G) ≥ 6. Proof of Theorem 1. Let v be an arbitrary vertex in G, with its incident edges labeled e1 , e2 , e3 , e4 in clockwise order and denote by Ai (i ∈ [4]) the set of the three edges adjacent to ei but not incident with v (as shown in Fig. 5). Now we begin with the following claim. Claim 4.1. Any partial coloring obtained by coloring one edge from each Ai can be extended a strong edge-coloring for G − v .
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Proof. For each i ∈ [4], let ai ∈ Ai be the edge chosen in the partial coloring. Define a partition V (G) = V1 ∪ V2 . . . ∪ Vs by letting Vi = {u | dist(u, v ) = i} for each i ∈ [s], where s is the largest distance from v to all the other vertices. Therefore this yields an ordering σ = (f1 , f2 , . . . , fm ) of all edges in G − {a1 , . . . , a4 , e1 , . . . , e4 } such that dist(v, fi ) ≤ dist(v, fj ) for any integers 1 ≤ i < j ≤ m. We now color the edges from fm back to f1 one by one in the ordering σ . Once we meet an edge e = xy with dist(x, v ) = l ≤ dist(y, v ). Let u be a neighbor of x which is in Vl−1 . Then at most one of the edges incident with u has been colored. Since |N c (e)| ≤ 24 − 3 = 21, we can always find a color available for e. □ Let Ai = {ai , bi , ci } and L(Ai ) = L(ai ) ∪ L(bi ) ∪ L(ci ) for each i ∈ [4]. Then we are ready to obtain a desired strong edge-coloring for G by precoloring one edge in each Ai . We consider the intersection relationship between L(Ai ) (i ∈ [4]).
⋃3
⋃4
Case 1. If L(Ai ) ∩ L(Aj ) = ∅ for any distinct i, j ∈ [4], then we have | i=1 L(Ai )| ≥ 4 × 22 > 3 × 22 ≥ | i=1 L(ei )|. So we ⋃2 ⋃4 ⋃3 ⋃4 obtain a color x1 ∈ i=1 L(Ai ) \ i=1 L(ei ) and assume x1 ∈ L(a1 ). Similarly, we have a color x2 ∈ i=2 L(Ai ) \ i=1 L(ei ) (assume x2 ∈ L′ (a2 )) and x3 ∈ L(A3 ) ∪ L(A4 ) \ L(e1 ) (assume x3 ∈ L(a3 )). By Claim 4.1, we precolor a1 , a2 , a3 with x1 , x2 , x3 respectively, and extend it to a strong edge-coloring of G − v . Then we have |L′ (e1 )| ≥ 4, |L′ (e2 )| ≥ 3, |L′ (e3 )| ≥ 2 and |L′ (e4 )| ≥ 1, so we can color the four edges in the order e4 , e3 , e2 , e1 .
⋂4
⋂4
Case 2. If i=1 L(Ai ) ̸ = ∅, we assume x ∈ i=1 L(Ai ). By Claim 4.1, we precolor ai with x for each i ∈ [4] (all edges ai are allowed to receive the same color because g(G) ≥ 6) and extend it to all edges of G − v . We observe that each ei satisfies |L(ei )| ≥ 4 and a desired strong edge-coloring easily follows.
⋂3
Case 3. If there exists a common color appeared in the list of three edge sets Ai , say x ∈ i=1 L(Ai ) \ L(A4 ), then we color ai (i ∈ [3]) with x. Since |L(A1 ) ∪ L(A4 )| ≥ 22 + 1 = |L(e1 )| + 1, it holds that L(A1 ) ∪ L(A4 ) \ L(e1 ) ̸ = ∅. If there is a color y ∈ L(A4 ) \ L(e1 ), then color a4 with y and extend it to all edges of G − v by Claim 4.1. It follows that |L′ (e1 )| ≥ 4, |L′ (ei )| ≥ 3 (2 ≤ i ≤ 4) and we color the four edges in the order e4 , e3 , e2 , e1 . Thus it remains to consider L(A4 ) = L(e1 ). Since x ∈ / L(A4 ) = L(e1 ), we color the remaining edges of G − v by Claim 4.1. It is easily observed that |L′ (e1 )| ≥ 4, ′ |L (ei )| ≥ 3 (2 ≤ i ≤ 4) and a desired strong edge-coloring follows. Case 4. If there is a color appeared in the lists of two edge sets Ai , say x ∈ L(A1 ) ∩ L(A2 ), and any three edge sets have no color in common. We begin the case with a claim as follows. Claim 4.2.
Under the assumption in Case 4, it holds that L(Ai ) ∩ L(Aj ) ⊆
⋂4
k=1
L(ek ) for any i, j with 1 ≤ i < j ≤ 4.
Proof. Suppose to the contrary. We further assume that x ∈ L(A1 ) ∩ L(A2 ) \ L(e1 ). Since x ∈ / L(A3 ) ∪ L(A4 ) and |L(A1 ) ∪ L(A3 )| ≥ 22 + 1 = |L(e2 )| + 1, there is a color y ∈ L(A1 ) ∪ L(A3 ) \ L(e2 ). We first assume that y ∈ L(A3 ) \ L(e2 ). By the assumption in Case 4, we have L(A2 ) ∪ L(A4 ) \ L(e2 ) ̸ = ∅. If there exists a color z ∈ L(A4 ) \ L(e2 ), then we assign a1 , a2 , a3 , a4 with x, x, y, z, respectively, and extend it to all edges of G − v by Claim 4.1. It is easily observed that |L′ (e1 )| ≥ 3, |L′ (e2 )| ≥ 4, |L′ (ei )| ≥ 2 (i = 3, 4) and a desired strong edge-coloring easily follows by coloring the four edges e4 , e3 , e1 , e2 in turn. If L(A4 ) = L(e2 ), then x ∈ / L(e2 ). We first assign a1 , a2 , a3 with colors x, x, y, respectively, and extend it to all edges of G − v by Claim 4.1. It follows that |L′ (e1 )| ≥ 3, |L′ (e2 )| ≥ 4, |L′ (ei )| ≥ 2 (i = 3, 4) and a desired coloring is obtained by coloring e4 , e3 , e1 , e2 in turn. It remains to consider the case L(A3 ) = L(e2 ). Then x ∈ / L(e2 ). Since L(A2 ) ∪ L(A4 ) \ L(e2 ) ̸= ∅, if there exists a color z ∈ L(A4 ) \ L(e2 ), then assign a1 , a2 , a4 with colors x, x, z, respectively, and extend it to all edges of G − v by Claim 4.1. It follows that |L′ (e1 )| ≥ 3, |L′ (e2 )| ≥ 4, |L′ (ei )| ≥ 2 (i = 3, 4) and a desired coloring is obtained by coloring the four edges e4 , e3 , e1 , e2 in turn. If L(A4 ) = L(e2 ), then we have L(A3 ) = L(A4 ). In this situation, we assign a3 , a4 with one color in L(A3 ) while assign a1 , a2 with color x, and extend it to all edges of G − v by Claim 4.1. It is easily observed that |L′ (e1 )| ≥ 4, |L′ (e2 )| ≥ 4, |L′ (ei )| ≥ 3 (i = 3, 4) and a desired coloring is obtained by coloring the four edges e4 , e3 , e2 , e1 in turn. □ ⋂4 ∑ Recall that L(A1 ) ∩ L(A2 ) ̸= ∅ and L(A1 ) ∩ L(A2 ) ∩ L(Ai ) = ∅ for each i = 3, 4. By Claim 4.2, we have i=1 L(ei ) ̸= ∅ and 1≤i
1≤i
⋃3 ⋃4 ⋃3 Since | i=1 L(ei )| < 3 × 22, there is a color y ∈ i=1 L(Ai ) \ i=1 L(ei ) and assume y ∈ L(a1 ). In this situation, by the same ⋃4 counting arguments, there is∑ also a color z ∈ i=2 L(Ai ) \ (L(e1 ) ∪ L(e2 )) and assume z ∈ L(a2 ). If |L(A3 ) ∪ L(A4 )| = 22, then we have L(A3 ) = L(A4 ) and 1≤i 22, and there is a color w ∈ L(A3 ) ∪ L(A4 ) \ L(e1 ) (assume w ∈ L(A3 )). Then we assign a1 , a2 , a3 with y, z, w , respectively, and extend it to all edges of G − v by Claim 4.1. It follows that |L′ (e1 )| ≥ 4, |L′ (e2 )| ≥ 3, |L′ (e3 )| ≥ 2, |L′ (e4 )| ≥ 1 and a desired coloring is obtained by coloring the four edges e4 , e3 , e2 , e1 in turn. This completes the proof of Theorem 1. □ 5. Concluding remarks In this paper we mainly study the list strong chromatic index for graphs with maximum degree 4. By the algebra method (Lemma 2.1) and Hall Theorem applied in the list context, we manage to follow the strategy of Cranston [4]. When
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preparing the paper, we realized that Huang, Santana and Yu [9] improved the upper bound for χs′ (G) with ∆(G) = 4 to 21 colors, one away from the conjectured 20. Hence it will be interesting and challenging to improve the upper bound 22 in Theorem 1 to 21. We will return to this topic in the near future. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The authors would like to thank Guanghui Wang for his helpful discussions and suggestions. This work was supported by the National Natural Science Foundation of China (11871311). Appendix %Matlab %input syms x1 x2 x3 x4 x5 x6 x7 x8 x9 %Lemma 2 . 4 Q=( x1 x2 ) ∗ (x1 x3 ) ∗ (x1 x4 ) ∗ (x1 x5 ) ∗ (x1 x6 ) ∗ (x1 x7 ) ∗ (x1 x9 ) ∗ (x2 x3 ) ∗ (x2 x4 ) ∗ (x2 x5 ) ∗ ( x2 x6 ) ∗ (x2 x7 ) ∗ (x2 x8 ) ∗ (x3 x4 ) ∗ (x3 x5 ) ∗ (x3 x6 ) ∗ (x3 x7 ) ∗ (x3 x8 ) ∗ (x3 x9 ) ∗ (x4 x5 ) ∗ ( x4 x7 ) ∗ (x4 x8 ) ∗ (x4 x9 ) ∗ (x5 x6 ) ∗ (x5 x8 ) ∗ (x5 x9 ) ∗ (x6 x7 ) ∗ (x7 x8 ) ∗ (x8 x9 ) ; C1= d i f f ( d i f f ( d i f f ( d i f f ( d i f f ( d i f f ( d i f f ( d i f f ( d i f f (Q , x1 , 3 ) , x2 , 4 ) , x3 , 5 ) , x4 , 4 ) , x5 , 4 ) , x6 , 2 ) , x7 , 3 ) , x8 , 2 ) , x9 , 2 ) / f a c t o r i a l ( 3 ) / f a c t o r i a l ( 4 ) / f a c t o r i a l ( 5 ) / f a c t o r i a l ( 4 ) / f a c t o r i a l (4) / f a c t o r i a l (2) / f a c t o r i a l (3) / f a c t o r i a l (2) / f a c t o r i a l (2) %output C1 = 1
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N. Alon, Combinatorial nullstellensatz, Comb. Probab. Comput. 8 (1999) 7–29. L.D. Andersen, The strong chromatic index of a cubic graph is at most 10, Discrete Math. 108 (1992) 231–252. M. Bonamy, T. Perrett, L. Postle, Colouring graphs with sparse neighbourhoods, Bounds and Applications, arXiv:1810.06704. D. Cranston, Strong edge-coloring of graphs with maximum degree 4 using 22 colors, Discrete Math. 306 (2006) 2772–2778. T. Dai, G. Wang, D. Yang, G. Yu, Strong choice number of subcubic graphs, Discrete Math. 341 (2018) 3434–3440. P. Erdős, Problems and results in combinatorial analysis ang graph theory, Discrete Math. 72 (1988) 81–92. P. Hall, On representatives of subsets, J. Lond. Math. Soc. 1 (1935) 26–30. P. Horák, H. Qing, W.T. Trotter, Induced matchings in cubic graphs, J. Graph Theory 17 (1993) 151–160. M. Huang, M. Santana, G. Yu, Strong chromatic index of graphs with maximum degree four, arXiv:1806.07012. T. Wang, Strong chromatic index of k-degenerate graphs, Discrete Math. 33 (2014) 17–19. D. Yang, X. Zhu, Strong chromatic index of sparse graphs, J. Graph Theory 83 (2016) 334–339. G. Yu, Strong edge-colorings for k-degenerate graphs, Graphs Combin. 31 (2015) 1815–1818.
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Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
List strong edge-coloring of graphs with maximum degree 4 Baochen Zhang a , Yulin Chang a , Jie Hu b , Meijie Ma c , Donglei Yang a ,
∗
a
School of Mathematics, Shandong University, Jinan, Shandong, 250100, China Laboratoire de Recherche en Informatique, Université Paris-Sud, Orsay Cedex 91405, France c School of Management Science and Engineering, Shandong Technology and Business University, Yantai, 264005, China b
article
info
Article history: Received 21 May 2018 Received in revised form 12 August 2019 Accepted 5 February 2020 Available online xxxx Keywords: List strong edge-coloring Combinatorial Nullstellensatz Hall’s Theorem
a b s t r a c t A strong edge-coloring of a graph G is an edge-coloring such that any two edges on a path of length three receive distinct colors. We denote the strong chromatic index by χs′ (G) which is the minimum number of colors that allow a strong edge-coloring of G. Erdős and Nešetřil conjectured in 1985 that the upper bound of χs′ (G) is 54 ∆2 when ∆ is even and 41 (5∆2 − 2∆ + 1) when ∆ is odd, where ∆ is the maximum degree of G. The conjecture is proved right when ∆ ≤ 3. The best known upper bound for ∆ = 4 is 22 (Cranston, 2006). In this paper we extend the result of Cranston to the list version, that is, we prove that when ∆ = 4, list strong chromatic index is at most 22. © 2020 Elsevier B.V. All rights reserved.
1. Introduction A strong edge-coloring is a proper edge-coloring such that no two edges on a path of length three have the same color. To be more precise, a strong k-edge-coloring of a graph G is a coloring φ : E(G) → [k] such that for any two edges e1 and e2 that are either adjacent to each other or adjacent to a common edge, φ (e1 ) ̸ = φ (e2 ). The strong chromatic index of G, denoted by χs′ (G), is the minimum positive integer k for which G has a strong k-edge-coloring. A graph G, together with a list assignment L = {L(e) : e ∈ E(G)}, is strongly L-edge-colorable if there exists a strong edge-coloring c for G such that c(e) ∈ L(e) for every edge e ∈ E(G). For a positive integer k, a graph G is strongly k-edgechoosable if G is strongly L-edge-colorable for every L with |L(e)| ≥ k for all e ∈ E(G). The strong choice number, denoted by χls′ (G), is the minimum positive integer k for which G is strongly k-edge-choosable. Given a graph G with the maximum degree ∆, Erdős and Nešetřil [6] conjectured that χs′ (G) is at most 45 ∆2 when ∆ is even and 41 (5∆2 − 2∆ + 1) when ∆ is odd. They also give a construction to show that if the conjecture is true, then the bound is tight. For graphs with ∆ = 3, the conjecture was proved by Andersen [2] and by Horák [8] independently. For ∆ = 4, the best known upper bound is 22, which is due to Cranston [4] . When ∆ is sufficiently large, Bonamy, Perrett and Postle [3] proved that χs′ (G) ≤ 1.835∆2 . As for k-degenerate graphs, Yu [12] proved that χs′ (G) ≤ (4k − 2)∆ − 2k2 + k + 1. More results can be found in [5,10,11]. In this paper, we mainly prove the following theorem which extends Cranston’s result [4] to the list version. Theorem 1.
Every graph G with maximum degree 4 is strongly 22-edge-choosable.
The rest of the paper is organized as follows. Section 2 introduces some basic notation and elementary tools. In Section 3, we explore some basic properties of the minimal counterexample to Theorem 1. We will complete the proof of Theorem 1 in Section 4. ∗ Corresponding author. E-mail address: [email protected] (D. Yang). https://doi.org/10.1016/j.disc.2020.111854 0012-365X/© 2020 Elsevier B.V. All rights reserved.
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Fig. 1. The neighborhood of e.
2. Preliminaries and notation Throughout this paper, we assume that G is connected and is allowed to contain loops and multiple edges. The girth of G, denoted by g(G), is the length of the shortest cycle in G. Let l-cycle be a cycle of length l. For any distinct vertices u and v , the distance dist(u, v ) is the length of the shortest path connecting u and v . For a vertex v and a cycle C , let dist(v, C ) := min{dist(v, u) | u ∈ V (C )}. In particular, for any two distinct edges e1 and e2 , let dist(e1 , e2 ) := min{distG (u, v ) | u ∈ e1 , v ∈ e2 }. The neighborhood NG (e) of an edge e is the set of edges e′ such that dist(e, e′ ) ≤ 1. If the graph G is clear from the context, then we will omit the subscript. A partial list edge coloring of G is an edge coloring c for a proper subgraph H of G with c(e) ∈ L(e) for each edge e ∈ E(H). Given a partial list edge coloring c and an edge e, a color in L(e) is available for e if the color is not used on any edges in N(e), and denote by L′ (e) the set of available colors for e. Let N c (e) be the set of edges in N(e) that have been colored in the coloring c. In particular, |N(e)| ≤ 24 for an edge e as shown in Fig. 1. In all figures of this paper, the black vertices have no other neighbors than those appeared while the white vertices may have other neighbors. One of the main tools we use is the Combinatorial Nullstellensatz due to Alon [1].
∑n
Lemma 2.1 ([1]). Let F be an arbitrary field, and let P = P(x1 , . . . , xn ) be a polynomial in F[x1 , . . . , xn ]. If deg(P) = i=1 ki , k k where each ki is a non-negative integer, and the coefficient of x11 · · · xnn in P is non-zero, then for any subsets S1 , . . . , Sn of F with |Si | > ki for each i ∈ [n], there exist s1 ∈ S1 , . . . , sn ∈ Sn such that P(s1 , . . . , sn ) ̸ = 0. Another tool is the well-known Hall’s Theorem. Lemma 2.2 ([7]). Let A1 , . . . , An be n subsets of a set U. Then there is a set {a1 , . . . , an } of distinct elements in U with ai ∈ Ai for each i ∈ [n] if and only if for any integer k with 1 ≤ k ≤ n and every subcollection {Ai1 , . . . , Aik } of subsets, we have |Ai1 ∪ · · · ∪ Aik | ≥ k. Suppose that we have a partial list coloring c of G with an edge set T uncolored. Let L′ (e) be the set of available colors for each e ∈ T . If c cannot ⋃ be extended to E(G) by coloring ⋃ each edge e ∈ T , then by Lemma 2.2, there exists a subset S ⊆ T such that |S | > | e∈S L′ (e)|. Define disc(S) = |S | − | e∈S L′ (e)| to be the discrepancy of S. The following lemma is a generalization of Lemma 1 in [4]. Lemma 2.3. Let T be the set of uncolored edges in a partially colored graph and S be a subset of T with maximum discrepancy. Then any list strong edge-coloring of S can be extended to the entire T . Proof. Assume there exists a list strong edge-coloring c of S that cannot be extended to T . Hence there exists ⋃ a subset ′ S ′ ⊆ T⋃ \S with positive discrepancy (under the coloring c). We claim that disc(S ∪ S ′ ) > disc(S). Let R = e∈S ∪S ′ L (e), ⋃ R1 = e∈S L′ (e), k = disc(S) and R2 = e∈S ′ L′′ (e), where L′′ (e) denotes the set of available colors for each e ∈ S ′ under the coloring c. Then |S | = k + |R1 | and |S ′ | ≥ 1 + |R2 |. Since S and S ′ are disjoint, we get
|S ∪ S ′ | = |S | + |S ′ | ≥ k + 1 + |R1 | + |R2 | > k + |R|. The latter inequality follows from the fact that |R| = |R1 ∪ R2 | ≤ |R1 |+|R2 |. Hence disc(S ∪ S ′ ) = |S ∪ S ′ |−|R| > k = disc(S). This contradicts the maximality of disc(S). Thus, any coloring of S can be extended to the entire T . □ In the end of this section, we prove a useful lemma for the main proof in Section 4. Lemma 2.4. Let G be a graph with maximum degree 4, and v be an arbitrary vertex in G, then G − v is strongly 21-edge-choosable. If C is a cycle of length at least 3 in G, then G − E(C ) is strongly 21-edge-choosable. Proof. We consider the first case G −v . Define a partition V (G) = V1 ∪ V2 . . .∪ Vs by letting Vi = {u | dist(u, v ) = i} for each i ∈ [s], where s is the largest distance from v to all the other vertices. Therefore this yields an ordering σ = (e1 , e2 , . . . , em )
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Fig. 2. The situation in the proof of Lemma 2.4.
of all edges of G − v such that dist(v, ei ) ≤ dist(v, ej ) for any integers 1 ≤ i < j ≤ m. We now color the edges from em back to e1 one by one in the ordering σ . Once we meet an edge e = xy with dist(x, v ) = l ≤ dist(y, v ). Let u be a neighbor of x which is in Vl−1 (see Fig. 2). Then none of the edges incident with u has been colored. Since |N c (e)| ≤ 20, we can always find a color available for e. We proceed the proof of the second part when the edges of the cycle C are left uncolored. Similarly, we define a partition V (G) = V1 ∪ V2 . . . ∪ Vs by letting Vi = {u | dist(u, C ) = i} for each i ∈ [s], which yields an ordering σ = (e1 , e2 , . . . , em′ ) of all edges of G − E(C ) such that dist(v, ei ) ≤ dist(v, ej ) for any integers 1 ≤ i < j ≤ m′ . We now color the edges from em′ back to e1 one by one in the ordering σ . By similar arguments as in the case G − v , we can greedily color every edge that is not incident with C . Let e be an edge incident to C . If |C | ≥ 4, then |N(e) ∩ E(C )| ≥ 4; so again |L′ (e)| ≥ 21−(24−4) = 1. If |C | = 3, then |N(e)| ≤ 23. The three uncolored edges of C imply that |L′ (e)| ≥ 21−(23−3) = 1. Thus we can easily obtain a desired strong edge-coloring for G. This completes the proof of Lemma 2.4. □ 3. Basic properties Let G be an edge-minimal counterexample to Theorem 1, which means there exists a list assignment L with |L(e)| = 22 for each e ∈ E(G) such that G is not strongly L-edge-colorable while any proper subgraph of G is strongly L-edge-colorable. In this section, we show that G is a simple 4-regular graph and g(G) ≥ 6. Claim 3.1.
G is 4-regular.
Proof. Suppose G is not 4-regular. Let v be a vertex of G with d(v ) ≤ 3. By Lemma 2.4, we have a partial list coloring c with all the incident edges e1 , . . . , et of v uncolored, where t ≤ 3. Hence, for each i ∈ [t ], |N c (ei )| ≤ 18, which means |L′ (ei )| ≥ 22 − 18 = 4. One can easily extend c to a strong edge-coloring for G, a contradiction. □ Claim 3.2.
G is simple.
Proof. Suppose G is not simple. If G has a loop e1 incident with a vertex v , then let e2 , e3 be the edges incident with v which are not loops. By Lemma 2.4, we have a partial list coloring c such that the three incident edges e1 , e2 , e3 of v are uncolored. Similarly we have |L′ (e1 )| ≥ 22 − 6 = 18, |L′ (e2 )| ≥ 22 − 15 = 7, |L′ (e3 )| ≥ 7, and one can easily extend c to the entire graph. Next we consider the other case when G has multiple edges. Let v be a vertex in a 2-cycle containing two edges e1 , e2 and e3 , e4 be the incident edges of v that are not in the 2-cycle. By Lemma 2.4, we have a partial list coloring c for G − v . Similarly we have |L′ (ei )| ≥ 22 − 14 = 8 for i = 1, 2 and |L′ (ej )| ≥ 22 − 17 = 5 for j = 3, 4. We can easily extend c to a strong edge-coloring for G. □ Claim 3.3.
G has no 3-cycle.
Proof. Suppose G has a 3-cycle C . By Lemma 2.4, we can color all edges except the edges of C . It follows that |L′ (e)| ≥ 22 − 18 = 4 for each e ∈ E(C ) and we can easily obtain a strong edge-coloring for G. □ Claim 3.4.
G has no 4-cycle.
Proof. Suppose G has a 4-cycle C with all edges labeled in Fig. 3(a). We call all the edges labeled ai and bi pendant edges. We say two pendant edges form an adjacent pair if they share an endpoint which is not in C . The only possibility is that a1 or b1 shares an endpoint with a3 or b3 (or similarly a2 or b2 shares an endpoint with a4 or b4 ). In particular, we call {a1 , b1 , a3 , b3 } (or {a2 , b2 , a4 , b4 }) a pack. If there are at least two adjacent pairs, then by Lemma 2.4, we have a partial list coloring c for G − E(C ). It follows that |L′ (ci )| ≥ 22 − 18 = 4 for each i ∈ [4]. So we can easily extend c to the entire graph. In this situation, there is at least one pack in which there is no adjacent pair, and each edge connecting two pendant edges in a pack is called a diagonal edge. Hence there are at most 4 diagonal edges for a pack (see Fig. 3(b)). We now claim that there are at most 3 diagonal edges in a pack. Indeed, if there exist 4 diagonal edges for a pack, say {a1 , b1 , a3 , b3 }, then the neighborhood of each diagonal edge has size at most 21. By Lemma 2.4, we can color all edges except the four diagonal
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Fig. 3. 4-cycle.
edges. Thus each uncolored diagonal edge e satisfies |L′ (e)| ≥ 22 − (21 − 3) = 4, and it follows that one can easily obtain a desired strong edge-coloring for G, a contradiction. By the minimality of G, G − V (C ) admits a strongly edge-coloring c. Then we consider the following two cases. (1) There is exactly one adjacent pair, say a2 and a4 . Then there exists an pair of edges in the pack {a1 , b1 , a3 , b3 }, say a1 and a3 , such that there is no diagonal edge connecting a1 , a3 . ′ Observe that |L′ (ci )| ≥ 11 for each i ∈ [4], |L′ (ai )| ≥ 7 and i = 1, 3. In addition, |L′ (b2 )| ≥ 8, |L′ (b4 )| ≥ ⋃|4L (bi )′ | ≥ 7 for ′ ′ ′ 8, |L (a2 )| ≥ 11 and |L (a4 )| ≥ 11. If there is a color x ∈ i=1 (L (ai ) ∪ L (bi )), but x ∈ / L(cj ) for some j ∈ [4] (suppose that color x ∈ L′ (a1 ) \ L′ (c1 )), then we can give a1 color x and color the remaining pendant edges by Lemma 2.4. After this, we have |L′ (ci )| ≥ 3 for i = 2, 3, 4 and |L′ (c1 )| ≥ 4, and we obtain a desired strong edge-coloring for G by coloring the four edges in the order c4 , c3 , c2 , c1 . Similarly, if |L′ (ci )| > 11 for some i ∈ [4], then we can also obtain a desired coloring by the same strategy as above. ⋃4 ′ ′ ′ ′ Thus it remains to consider the case when i=1 (L (ai ) ∪ L (bi )) ⊆ L (cj ) and |L (cj )| = 11 for each j ∈ [4]. Since ′ ′ ′ ′ |L (a1 )| ≥ 7 and |L (a3 )| ≥ 7, there are at least 3 colors in L (a1 ) ∩ L (a3 ). We assign a1 and a3 with one color y ∈ L′ (a1 ) ∩ L′ (a3 ) and color the remaining pendant edges by Lemma 2.4. After this, it holds that |L′ (ci )| ≥ 4 for each i ∈ [4]. Thus we have a desired strong edge-coloring for G. (2) There is no adjacent pair. Since |L′ (ai )| ≥ 7, |L′ (bi )| ≥ 7 and |L′ (ci )| ≥ 10 for each i ∈ [4]. By Lemma 2.2 applied to the 12 uncolored edges, if we cannot assign a distinct color to each edge, then there is a subset S of the 12 uncolored edges with maximum positive discrepancy. We may assume that S contains an edge ci for some i ∈ [4], otherwise we can color S by Lemma 2.4, then extend the partial coloring to the remaining uncolored edges by Lemma 2.3. Since disc(S) > 0 and |L′ (ci )| ≥ 10, it follows that |S | is 11 or 12. If |S | = 12, then the two packs {a1 , b1 , a3 , b3 } and {a2 , b2 , a4 , b4 } are all in S. Recall that each pack has a pair of nonadjacent edges with no diagonal edge between them, suppose that the two pairs are (a1 , a3 ) and (a2 , b4 ). Since |L′ (a1 )| ≥ 7, |L′ (a3 )| ≥ 7, and |L′ (a1 ) ∪ L′ (a3 )| ≤ |S | − disc(S) ≤ 11, we have |L′ (a1 ) ∩ L′ (a3 )| ≥ 3. Similarly, we have |L′ (a2 ) ∩ L′ (b4 )| ≥ 3. Then we choose two distinct colors x, y with x ∈ L′ (a1 ) ∩ L′ (a3 ), y ∈ L′ (a2 ) ∩ L′ (b4 ) and assign a1 , a3 with color x while assign a2 and b4 with color y. After coloring the remaining pendant edges by Lemma 2.4, we have |L′ (ci )| ≥ 4 for each i ∈ [4] and we are done. If |S | = 11, then there is an uncolored edge e such that e ∈ / S. We may further assume that e is a pendant edge, otherwise we can color the 12 edges by the same argument as the case |S | = 12. Suppose that e = a1 . Since the pack {a2 , b2 , a4 , b4 } has a pair of nonadjacent edges with no diagonal edge between them, say (a2 , a4 ). Then again we can assign a2 , a4 with the same color and color the remaining pendant edges in S by Lemma 2.4. In this situation, a1 is uncolored and the resulting available color set L′′ (ci ) for each ci satisfies |L′′ (ci )| ≥ 10 − 7 + 1 = 4. Then one can easily obtain a partial coloring for S and extend it to a final coloring for the 12 edges by Lemma 2.3. This completes the proof of Claim 3.4. □ Claim 3.5.
G has no 5-cycle.
Proof. Suppose G has a 5-cycle C shown in Fig. 4(a). We also refer to the edges labeled by ai and bi as pendant edges. We claim that at least one of a4 and b4 is not in the neighborhood of b2 , otherwise we have a 4-cycle. Similarly, we assume that there is no edge between the following pairs: (b1 , b3 ), (b2 , b5 ) and (b5 , b3 ). We color all edges except five edges on C by Lemma 2.4 and then erase the colors of b2 , b3 , b4 , b5 . The resulting coloring implies that |L′ (ci )| ≥ 5 for i = 1, 2, 4, 5,
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Fig. 4. 5-cycle.
Fig. 5. Proof of Theorem 1: Final analysis.
|L′ (c3 )| ≥ 6, |L′ (b2 )| ≥ 3, |L′ (b5 )| ≥ 3 and |L′ (b3 )| ≥ 4, |L′ (b4 )| ≥ 4. We relabel the edges as shown in Fig. 4(b). Hence we will finish the coloring if there exist si ∈ L′ (xi ) for each i ∈ [9] such that P(s1 , . . . , sn ) ̸ = 0 where ∏ ∏ ∏ P(x1 , x2 , . . . , x9 ) = (x1 − xi ) (x2 − xj ) (x3 − xk ) 2≤i≤7
3≤j≤8
4≤k≤9
(x1 − x9 )(x4 − x5 )(x4 − x7 )(x4 − x8 )(x4 − x9 ) (x5 − x6 )(x5 − x8 )(x5 − x9 )(x6 − x7 )(x7 − x8 ) (x8 − x9 ). We use MATLAB to calculate the coefficients of specific monomials. The codes are listed in the last section. By MATLAB, we obtain a coefficient cP (x31 x42 x53 x44 x45 x26 x37 x28 x29 ) = −1 ̸ = 0. Since deg(P) = 29, Lemma 2.1 implies that there exist si ∈ L′ (xi ) for each i ∈ [9] such that P(s1 , . . . , s9 ) ̸ = 0. Then we obtain a strong edge-coloring for G by coloring x1 , . . . , x9 with s1 , . . . , s9 respectively, a contradiction. □ 4. Final analyis In this section, we complete the proof of Theorem 1. Recall that G is a minimal counterexample to Theorem 1 and by Claim 3.1–3.5, G is a simple 4-regular graph with g(G) ≥ 6. Proof of Theorem 1. Let v be an arbitrary vertex in G, with its incident edges labeled e1 , e2 , e3 , e4 in clockwise order and denote by Ai (i ∈ [4]) the set of the three edges adjacent to ei but not incident with v (as shown in Fig. 5). Now we begin with the following claim. Claim 4.1. Any partial coloring obtained by coloring one edge from each Ai can be extended a strong edge-coloring for G − v .
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B. Zhang, Y. Chang, J. Hu et al. / Discrete Mathematics 343 (2020) 111854
Proof. For each i ∈ [4], let ai ∈ Ai be the edge chosen in the partial coloring. Define a partition V (G) = V1 ∪ V2 . . . ∪ Vs by letting Vi = {u | dist(u, v ) = i} for each i ∈ [s], where s is the largest distance from v to all the other vertices. Therefore this yields an ordering σ = (f1 , f2 , . . . , fm ) of all edges in G − {a1 , . . . , a4 , e1 , . . . , e4 } such that dist(v, fi ) ≤ dist(v, fj ) for any integers 1 ≤ i < j ≤ m. We now color the edges from fm back to f1 one by one in the ordering σ . Once we meet an edge e = xy with dist(x, v ) = l ≤ dist(y, v ). Let u be a neighbor of x which is in Vl−1 . Then at most one of the edges incident with u has been colored. Since |N c (e)| ≤ 24 − 3 = 21, we can always find a color available for e. □ Let Ai = {ai , bi , ci } and L(Ai ) = L(ai ) ∪ L(bi ) ∪ L(ci ) for each i ∈ [4]. Then we are ready to obtain a desired strong edge-coloring for G by precoloring one edge in each Ai . We consider the intersection relationship between L(Ai ) (i ∈ [4]).
⋃3
⋃4
Case 1. If L(Ai ) ∩ L(Aj ) = ∅ for any distinct i, j ∈ [4], then we have | i=1 L(Ai )| ≥ 4 × 22 > 3 × 22 ≥ | i=1 L(ei )|. So we ⋃2 ⋃4 ⋃3 ⋃4 obtain a color x1 ∈ i=1 L(Ai ) \ i=1 L(ei ) and assume x1 ∈ L(a1 ). Similarly, we have a color x2 ∈ i=2 L(Ai ) \ i=1 L(ei ) (assume x2 ∈ L′ (a2 )) and x3 ∈ L(A3 ) ∪ L(A4 ) \ L(e1 ) (assume x3 ∈ L(a3 )). By Claim 4.1, we precolor a1 , a2 , a3 with x1 , x2 , x3 respectively, and extend it to a strong edge-coloring of G − v . Then we have |L′ (e1 )| ≥ 4, |L′ (e2 )| ≥ 3, |L′ (e3 )| ≥ 2 and |L′ (e4 )| ≥ 1, so we can color the four edges in the order e4 , e3 , e2 , e1 .
⋂4
⋂4
Case 2. If i=1 L(Ai ) ̸ = ∅, we assume x ∈ i=1 L(Ai ). By Claim 4.1, we precolor ai with x for each i ∈ [4] (all edges ai are allowed to receive the same color because g(G) ≥ 6) and extend it to all edges of G − v . We observe that each ei satisfies |L(ei )| ≥ 4 and a desired strong edge-coloring easily follows.
⋂3
Case 3. If there exists a common color appeared in the list of three edge sets Ai , say x ∈ i=1 L(Ai ) \ L(A4 ), then we color ai (i ∈ [3]) with x. Since |L(A1 ) ∪ L(A4 )| ≥ 22 + 1 = |L(e1 )| + 1, it holds that L(A1 ) ∪ L(A4 ) \ L(e1 ) ̸ = ∅. If there is a color y ∈ L(A4 ) \ L(e1 ), then color a4 with y and extend it to all edges of G − v by Claim 4.1. It follows that |L′ (e1 )| ≥ 4, |L′ (ei )| ≥ 3 (2 ≤ i ≤ 4) and we color the four edges in the order e4 , e3 , e2 , e1 . Thus it remains to consider L(A4 ) = L(e1 ). Since x ∈ / L(A4 ) = L(e1 ), we color the remaining edges of G − v by Claim 4.1. It is easily observed that |L′ (e1 )| ≥ 4, ′ |L (ei )| ≥ 3 (2 ≤ i ≤ 4) and a desired strong edge-coloring follows. Case 4. If there is a color appeared in the lists of two edge sets Ai , say x ∈ L(A1 ) ∩ L(A2 ), and any three edge sets have no color in common. We begin the case with a claim as follows. Claim 4.2.
Under the assumption in Case 4, it holds that L(Ai ) ∩ L(Aj ) ⊆
⋂4
k=1
L(ek ) for any i, j with 1 ≤ i < j ≤ 4.
Proof. Suppose to the contrary. We further assume that x ∈ L(A1 ) ∩ L(A2 ) \ L(e1 ). Since x ∈ / L(A3 ) ∪ L(A4 ) and |L(A1 ) ∪ L(A3 )| ≥ 22 + 1 = |L(e2 )| + 1, there is a color y ∈ L(A1 ) ∪ L(A3 ) \ L(e2 ). We first assume that y ∈ L(A3 ) \ L(e2 ). By the assumption in Case 4, we have L(A2 ) ∪ L(A4 ) \ L(e2 ) ̸ = ∅. If there exists a color z ∈ L(A4 ) \ L(e2 ), then we assign a1 , a2 , a3 , a4 with x, x, y, z, respectively, and extend it to all edges of G − v by Claim 4.1. It is easily observed that |L′ (e1 )| ≥ 3, |L′ (e2 )| ≥ 4, |L′ (ei )| ≥ 2 (i = 3, 4) and a desired strong edge-coloring easily follows by coloring the four edges e4 , e3 , e1 , e2 in turn. If L(A4 ) = L(e2 ), then x ∈ / L(e2 ). We first assign a1 , a2 , a3 with colors x, x, y, respectively, and extend it to all edges of G − v by Claim 4.1. It follows that |L′ (e1 )| ≥ 3, |L′ (e2 )| ≥ 4, |L′ (ei )| ≥ 2 (i = 3, 4) and a desired coloring is obtained by coloring e4 , e3 , e1 , e2 in turn. It remains to consider the case L(A3 ) = L(e2 ). Then x ∈ / L(e2 ). Since L(A2 ) ∪ L(A4 ) \ L(e2 ) ̸= ∅, if there exists a color z ∈ L(A4 ) \ L(e2 ), then assign a1 , a2 , a4 with colors x, x, z, respectively, and extend it to all edges of G − v by Claim 4.1. It follows that |L′ (e1 )| ≥ 3, |L′ (e2 )| ≥ 4, |L′ (ei )| ≥ 2 (i = 3, 4) and a desired coloring is obtained by coloring the four edges e4 , e3 , e1 , e2 in turn. If L(A4 ) = L(e2 ), then we have L(A3 ) = L(A4 ). In this situation, we assign a3 , a4 with one color in L(A3 ) while assign a1 , a2 with color x, and extend it to all edges of G − v by Claim 4.1. It is easily observed that |L′ (e1 )| ≥ 4, |L′ (e2 )| ≥ 4, |L′ (ei )| ≥ 3 (i = 3, 4) and a desired coloring is obtained by coloring the four edges e4 , e3 , e2 , e1 in turn. □ ⋂4 ∑ Recall that L(A1 ) ∩ L(A2 ) ̸= ∅ and L(A1 ) ∩ L(A2 ) ∩ L(Ai ) = ∅ for each i = 3, 4. By Claim 4.2, we have i=1 L(ei ) ̸= ∅ and 1≤i
1≤i
⋃3 ⋃4 ⋃3 Since | i=1 L(ei )| < 3 × 22, there is a color y ∈ i=1 L(Ai ) \ i=1 L(ei ) and assume y ∈ L(a1 ). In this situation, by the same ⋃4 counting arguments, there is∑ also a color z ∈ i=2 L(Ai ) \ (L(e1 ) ∪ L(e2 )) and assume z ∈ L(a2 ). If |L(A3 ) ∪ L(A4 )| = 22, then we have L(A3 ) = L(A4 ) and 1≤i
B. Zhang, Y. Chang, J. Hu et al. / Discrete Mathematics 343 (2020) 111854
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preparing the paper, we realized that Huang, Santana and Yu [9] improved the upper bound for χs′ (G) with ∆(G) = 4 to 21 colors, one away from the conjectured 20. Hence it will be interesting and challenging to improve the upper bound 22 in Theorem 1 to 21. We will return to this topic in the near future. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The authors would like to thank Guanghui Wang for his helpful discussions and suggestions. This work was supported by the National Natural Science Foundation of China (11871311). Appendix %Matlab %input syms x1 x2 x3 x4 x5 x6 x7 x8 x9 %Lemma 2 . 4 Q=( x1 x2 ) ∗ (x1 x3 ) ∗ (x1 x4 ) ∗ (x1 x5 ) ∗ (x1 x6 ) ∗ (x1 x7 ) ∗ (x1 x9 ) ∗ (x2 x3 ) ∗ (x2 x4 ) ∗ (x2 x5 ) ∗ ( x2 x6 ) ∗ (x2 x7 ) ∗ (x2 x8 ) ∗ (x3 x4 ) ∗ (x3 x5 ) ∗ (x3 x6 ) ∗ (x3 x7 ) ∗ (x3 x8 ) ∗ (x3 x9 ) ∗ (x4 x5 ) ∗ ( x4 x7 ) ∗ (x4 x8 ) ∗ (x4 x9 ) ∗ (x5 x6 ) ∗ (x5 x8 ) ∗ (x5 x9 ) ∗ (x6 x7 ) ∗ (x7 x8 ) ∗ (x8 x9 ) ; C1= d i f f ( d i f f ( d i f f ( d i f f ( d i f f ( d i f f ( d i f f ( d i f f ( d i f f (Q , x1 , 3 ) , x2 , 4 ) , x3 , 5 ) , x4 , 4 ) , x5 , 4 ) , x6 , 2 ) , x7 , 3 ) , x8 , 2 ) , x9 , 2 ) / f a c t o r i a l ( 3 ) / f a c t o r i a l ( 4 ) / f a c t o r i a l ( 5 ) / f a c t o r i a l ( 4 ) / f a c t o r i a l (4) / f a c t o r i a l (2) / f a c t o r i a l (3) / f a c t o r i a l (2) / f a c t o r i a l (2) %output C1 = 1
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