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Edge-coloring of plane multigraphs with many colors on facial cycles
Discrete Applied Mathematics xxx (xxxx) xxx
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Edge-coloring of plane multigraphs with many colors on facial cycles✩ ∗
Július Czap a , , Stanislav Jendrol’ b , Juraj Valiska b a
Department of Applied Mathematics and Business Informatics, Faculty of Economics, Technical University of Košice, Němcovej 32, 040 01 Košice, Slovakia b Institute of Mathematics, P. J. Šafárik University, Jesenná 5, 040 01 Košice, Slovakia
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Article history: Received 13 May 2019 Received in revised form 24 October 2019 Accepted 4 November 2019 Available online xxxx Keywords: Plane graph Edge-coloring
a b s t r a c t For a fixed positive integer p, a coloring of the edges of a multigraph 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 coloring of G is the p-arboricity of G. From a result of Bartnicki et al. (2019) it follows that there are planar graphs with unbounded p-arboricity. In this note we relax the definition of p-arboricity for plane multigraphs in sense that the requirement is not for all cycles but only for facial cycles and show that the smallest number of colors needed for such a coloring is a constant (depending on p only). © 2019 Elsevier B.V. All rights reserved.
1. Introduction All graphs considered in this paper are loopless, parallel edges are allowed provided that it is not stated otherwise. We use standard graph theory terminology according to [4]. However, the most frequent notions of the paper are defined through it. A plane graph is a particular drawing of a planar graph in the Euclidean plane such that no edges intersect. Let G be a connected plane graph with vertex set V (G), edge set E(G), and face set F (G). The boundary of a face f is the boundary in the usual topological sense. It is the collection of all edges and vertices contained in the closure of f that can be organized into a closed walk in G traversing along a simple closed curve lying just inside the face f . This closed walk is unique up to the choice of initial vertex and direction, and is called the boundary walk of the face f (see [13], p. 101). The size of a face f , denoted by deg(f ), is the length of its boundary walk. A k-face is a face of size k. Two vertices (two edges) are adjacent if they are connected by an edge (have a common endvertex). A vertex and an edge are incident if the vertex is an endvertex of the edge. A vertex (or an edge) and a face are incident if the vertex (or the edge) lies on the boundary of the face. Two edges of a plane graph G are facially adjacent if they are consecutive on the boundary walk of a face of G. An edge-coloring of a graph G is an assignment of colors to the edges, one color to each edge. An edge-coloring c of a graph G is proper if for any two adjacent edges e1 and e2 of G, c(e1 ) ̸ = c(e2 ) holds. A facial edge-coloring c of a plane graph G is an edge-coloring such that for any two facially adjacent edges e1 and e2 of G, c(e1 ) ̸ = c(e2 ) holds. Observe that this coloring need not to be proper in a usual sense. We require only that facially ✩ An extended abstract of the research presented in this paper has been accepted at the European Conference on Combinatorics, Graph Theory and Applications (EUROCOMB 2019). ∗ Corresponding author. E-mail addresses: [email protected] (J. Czap), [email protected] (S. Jendrol’), [email protected] (J. Valiska). https://doi.org/10.1016/j.dam.2019.11.003 0166-218X/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: J. Czap, S. Jendrol’ and J. Valiska, Edge-coloring of plane multigraphs with many colors on facial cycles, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.003.
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adjacent edges must receive different colors. There are many results concerning different types of facial edge-colorings of plane graphs, see e.g. recent surveys [6] and [7]. A proper edge-coloring of a simple graph G is r-acyclic if every cycle C contained in G is colored with at least min{|C |, r } colors. The r-acyclic chromatic index a′r (G) of G is the minimum number of colors needed for an r-acyclic edge-coloring. Clearly, at least ∆ colors are required for any r-acyclic edge-coloring of a graph with maximum degree ∆. A 2-acyclic edgecoloring coincides with a proper edge-coloring, thus by the well-known Vizing’s theorem [17], ∆(G) ≤ a′2 (G) ≤ ∆(G) + 1 holds for every (simple) graph G. Alon, McDiarmid, and Reed [1] showed that the 3-acyclic chromatic index is linear in ∆. So far, the best upper bound for a′3 (G) is ⌈3.74(∆(G) − 1)⌉ + 1, which was obtained by Giotis et al. [11]. 3-acyclic edge-coloring has been deeply studied for planar graphs. The best known upper bound for this class of graphs is ∆ + 6, see [18]. For r ≥ 4 the situation is different, in this case the r-acyclic chromatic index may not be linear. In general, for every r r ≥ 4 there exist graphs with r-acyclic chromatic index at least cr · ∆⌊ 2 ⌋ , where cr is a constant (it depends only on r), see [12]. On the other hand, if the girth of a graph is sufficiently large, then its r-acyclic chromatic index is linear in ∆ if r is fixed. Gerke and Raemy [10] proved that for every graph G with girth at least 3(r − 1)∆(G) it holds that a′r (G) ≤ 6(r − 1)∆(G). In their theorem the girth is related to both r and ∆. Ding, Wang, and Wu [8] also proved a linear upper bound in terms of girth and maximum degree, but in their result the girth depends only on r. They proved that every graph with girth at least 2r − 1 admits an r-acyclic edge-coloring with at most (9r − 7)∆ + 10r − 12 colors. There is only one known result on r-acyclic edge-coloring of planar graphs with r ≥ 4. Zhang et al. [19] showed that a′4 (G) ≤ 37∆(G) for every planar graph G. The improper version of this problem is known as a generalized arboricity in the literature. The arboricity of a graph G is the minimum number of colors needed to color the edges of G so that every color class induces a forest. Therefore, if we require that every cycle gets at least two colors, then the minimum number of colors needed is the arboricity of G, and its determination is solved by the well-known Nash-Williams’ formula [14]. This concept was generalized by Nešetřil, Ossona de Mendez, and Zhu [15] in 2014. They introduced the p-arboricity of a graph G as the minimum number of colors needed to color the edges of G in such a way that every cycle C gets at least min{|C |, p + 1} colors. Obviously, the 1-arboricity is the classical arboricity. Bartnicki et al. [3] showed that the p-arboricity cannot be generally bounded by any constant (depending on p only) even for planar graphs, for all p ≥ 2. In this paper we relax the definition of p-arboricity for plane graphs in sense that the requirement is not for all cycles but only for facial cycles. We introduce the following parameter: For a fixed positive integer p, a facial edge-coloring of a plane graph G is called facial p-coloring with respect to faces, if there are at least min{|E(f )|, p} colors on the boundary of each face f , where E(f ) denotes the set of edges incident with the face f in G. The facial p-arboricity of a plane graph G with respect to faces, denoted by arbrf (p, G), is the smallest number of colors needed for a facial p-coloring with respect to faces. Our main result is that the facial p-arboricity of a connected plane graph G can be bounded by a constant depending on p only, which is a significant difference between the facial p-arboricity and the p-arboricity. 2. p-arboricity of planar graphs with large girth Bartnicki et al. [3] proved that there are planar graphs with unbounded p-arboricity, for all p ≥ 2. On the other hand, they proved that planar graphs with large girth have p-arboricity equal to p + 1. Theorem 1 ([3]). The p-arboricity of every planar graph of girth at least 2p+1 is p + 1, for all p ≥ 2. In the following we improve this result. In our theorem, the girth is bounded by a linear function of p. Theorem 2.
The p-arboricity of every planar graph of girth at least 5p + 1 is p + 1, for all p ≥ 2.
Proof. Suppose that there is a counterexample to Theorem 2. Let H be a counterexample with minimum number of vertices. The minimum degree of H is at least 2. To see this, suppose that H has a vertex v of degree 1. The graph H − v has fewer vertices than H so its p-arboricity is p + 1. The coloring of H − v can be extended to a coloring of H in such a way that we color the edge incident with v with an arbitrary color. Now we show that H has a path on p + 1 edges such that all of its internal vertices have degree 2 in H. If we iron the vertices of degree 2 in H, then we obtain a plane graph H ′ with minimum degree at least three. By Euler’s formula, |V (H ′ )| − |E(H ′ )| + |F (H ′ )| = 2. Using the Handshaking Lemma
∑
∑
deg(v ) = 2|E(H ′ )| =
v∈V (H ′ )
deg(f ),
f ∈F (H ′ )
we have
∑
(2 deg(v ) − 6) +
v∈V (H ′ )
∑
(deg(f ) − 6) = −12.
f ∈F (H ′ )
Please cite this article as: J. Czap, S. Jendrol’ and J. Valiska, Edge-coloring of plane multigraphs with many colors on facial cycles, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.003.
J. Czap, S. Jendrol’ and J. Valiska / Discrete Applied Mathematics xxx (xxxx) xxx
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Therefore, H ′ has a face f of size at most 5. Since the corresponding face in H has at least 5p + 1 edges, at least one edge of f is subdivided at least p times in H. So let P = v0 v1 . . . vp+1 be an induced path in H. By the minimality of the graph H, the p-arboricity of H − {v1 , . . . , vp } equals p + 1. Now color the edge vi vi+1 of P with color i + 1, for i = 0, . . . , p. Clearly, if a cycle of H contains the edges of P, then there are p + 1 colors on its edges. On the other hand, if a cycle C does not contain the edges of P, then C belongs to H − {v1 , . . . , vp }, consequently all p + 1 colors appear on C . □ 3. Facial p-arboricity of plane graphs with respect to faces Given a graph G and one of its edges e = uv , the contraction of e consists of replacing u and v by a new vertex adjacent to all the former neighbors of u and v , and removing the loop corresponding to the edge e. We denote the resulting graph by G/e. The dual G∗ of a plane graph G is obtained as follows: Corresponding to each face f of G there is a vertex f ∗ of G∗ , and corresponding to each edge e of G there is an edge e∗ of G∗ ; two vertices f ∗ and g ∗ are joined by the edge e∗ in G∗ if and only if their corresponding faces f and g are separated by the edge e in G (an edge separates the faces incident with it). The simplified medial M(G) of a plane graph G can be obtained as follows: Corresponding to each edge e of G there is a vertex m(e) of M(G); two vertices m(e1 ) and m(e2 ) are joined by an edge in M(G) if and only if their corresponding edges e1 and e2 are facially adjacent in G. It is known that every connected plane graph has a facial edge-coloring with at most 4 colors, moreover this bound is tight, see e.g. [9]. From the fact that in every facial edge-coloring every face has at least two colors on the boundary it follows that arbrf (2, G) ≤ 4 for every plane graph G. Theorem 3.
If p ≥ 3 is an integer and G a connected plane graph, then arbrf (p, G) ≤
3 p. 2
Moreover, this bound is tight.
Proof. Let p = 3. Let M(G) be the simplified medial of G. For every face f of M(G) of size at least four with boundary walk v1 v2 . . . vk we insert the edge v1 v3 into f . In such a way we obtain a new plane graph H. By the Four Color Theorem [2], H has a proper vertex-coloring with at most four colors. This vertex-coloring of H induces a facial edge-coloring of G. Since every 3-cycle in H uses three different colors, every face of G has at least three colors on the boundary. Now assume that p ≥ 4. Suppose there is a counterexample to Theorem 3. Let H be a counterexample with minimum number of vertices, then minimum number of edges. First we prove several structural properties of H. Claim 1.
H does not contain any cut-edge.
Proof. Suppose that H has a cut-edge e. Let fe be the face incident with e in H. By the minimality of H, the graph H /e has a facial edge-coloring with at most 23 p colors such that at least min{|E(f )|, p} colors appear on the boundary of each face f of H /e. Let fe′ be the face of H /e corresponding to fe in H. If less than p colors are used on the boundary of fe′ , then we color the edge e with a color which does not appear on fe′ . Otherwise we color e with a color that is not used on the edges facially adjacent to e. Such a color exists, since 32 p ≥ 6 and every edge is facially adjacent with at most four edges. □ Claim 2.
H does not contain any 2-face.
Proof. Suppose that H contains a 2-face f bounded with edges e1 = uv and e2 = uv . Let fe1 be the second face incident with e1 in H. The graph H −e1 has fewer edges than H, so it has a suitable coloring. Let fe′1 be the face of H −e1 corresponding to fe1 in H. If less than p + 1 colors are used on the boundary of fe′1 , then we color the edge e1 with a color which does not appear on fe′1 . Otherwise we color e1 with a color that is not used on the edges facially adjacent to e1 . □ Claim 3.
H does not contain parallel edges.
Proof. Suppose that H contains parallel edges e1 = uv and e2 = uv . By Claim 2, no two parallel edges form a 2-face, therefore there are at least one vertex inside and at least one vertex outside the cycle C formed by e1 and e2 . Let H1 be the graph obtained from H by deleting the vertices and edges outside C , and H2 be the graph obtained from H by deleting the vertices and edges inside C . The graphs H1 and H2 have fewer vertices than H, therefore each of them has a suitable coloring. Clearly, the edges incident with C are colored with two colors in both graphs H1 and H2 . Hence, by renaming the colors if necessary, the colorings of H1 and H2 induce a suitable coloring of H. □ Claim 4.
H does not contain any face of size at least p + 1.
Proof. Suppose that H contains a face f1 of size at least p + 1. Consider an edge e that f1 shares with a face f2 . By the minimality of H, the graph H /e has a facial edge-coloring with at most 23 p colors such that at least min{|E(f )|, p} colors appear on the boundary of each face f of H /e (the graph H /e has no loops by Claim 3). We extend the coloring of H /e to Please cite this article as: J. Czap, S. Jendrol’ and J. Valiska, Edge-coloring of plane multigraphs with many colors on facial cycles, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.003.
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a coloring of H in the following way. Let fi′ be the face of H /e corresponding to fi in H, for i = 1, 2. Observe that, there are at least p colors on f1′ , since its size is at least p. If at least p colors appear on f2′ , then we color e with a color that is not used on the edges facially adjacent to e. If less than p colors appear on f2′ , then we color e with a color that is not used on f2′ and on the edges facially adjacent to e. Such a color exists, since p + 2 ≤ 23 p for p ≥ 4. □ From Claims 1–4 it follows that the graph H is 2-edge-connected and each of its faces has size at most p. Consequently, its dual H ∗ is loopless and has maximum degree at most p. Since each face of H has size at most p, in any facial p-coloring with respect to faces, the edges bounding every face are colored distinctly. Such a coloring of H induces a proper edgecoloring of H ∗ and vice versa. Shannon [16] proved that every graph with maximum degree p has a proper edge-coloring with at most 32 p colors. Consequently, H has a suitable coloring, i.e. there is no counterexample to Theorem 3. p To see that the bound is tight, it suffices to paste together two p-cycles on 2 edges (if p is even). In this way we obtain a plane graph G which has three faces of size p. Thus, p different colors must appear on every face. Moreover, any two edges of G are incident with a common face, therefore no two edges are colored with the same color. The graph G has 23 p edges, hence arbrf (p, G) = 23 p. p−1 If p is odd, then it is sufficient to paste together a p-cycle and a (p − 1)-cycle on 2 edges. □ Since there are 2-edge-connected plane graphs G with arbrf (p, G) = 23 p, in the following we consider 3-edge-connected plane graphs and show that the bound 32 p can be significantly improved for this class of graphs. By Menger’s theorem, a graph is 3-edge-connected if and only if every pair of vertices is joined by at least 3 pairwise edge-disjoint paths. In the proofs of Theorems 4 and 5 we will use the fact that contracting an edge preserves edge-connectivity, see [4] p. 218. Proposition 1. If G is a k-edge-connected graph on at least three vertices, and e is any edge of G, then G/e is k-edge-connected. From Theorem 3 it follows that arbrf (3, G) ≤ 4. Therefore in the following we consider p ≥ 4. Theorem 4.
If p ≥ 4 is an integer and G a 3-edge-connected plane graph, then arbrf (p, G) ≤ p + 2.
Proof. Suppose there is a counterexample to Theorem 4. Let H be a counterexample with minimum number of vertices, then minimum number of edges. First we prove that H has no parallel edges. Suppose that H contains a 2-face f bounded with edges e1 = uv and e2 = uv . We distinguish three cases according to the number of uv edges in H. Case 1. First assume that there are at least four uv edges in H. In this case H − e1 is 3-edge-connected and has fewer edges than G, thus it has a suitable coloring. Let fe1 be the second face incident with e1 in H, and let fe′1 be the face of H − e1 corresponding to fe1 in H. We extend a suitable coloring of H − e1 to a suitable coloring of H as follows. If less than p + 1 colors are used on the boundary of fe′1 , then we color the edge e1 with a color which does not appear on fe′1 . Otherwise we color e1 with a color that is not used on the edges facially adjacent to e1 . Case 2. Now assume that there are two uv edges in H. Let H ′ be the graph obtained from H by removing the two uv edges and identifying the vertices u and v , i.e. H ′ = (H − e1 )/e2 . Clearly, H ′ is 3-edge-connected. Let fei ̸ = f be the face incident with ei in H, and let fe′i be the face of H ′ corresponding to fei in H, i = 1, 2. We extend a suitable coloring of H ′ in the following way. First we color the edge e1 . If less than p colors are used on the boundary of fe′1 , then we color the edge e1 with a color which does not appear on fe′1 . Otherwise we color e1 with a color that is not used on the edges facially adjacent to e1 . Now we color e2 . If less than p colors are used on the boundary of fe′2 , then we color the edge e2 with a color which does not appear on fe′2 and on the edge e1 . Otherwise we color e2 with a color that is not used on the edges facially adjacent to e2 . Case 3. Finally, assume that there are three uv edges e1 , e2 , e3 in H. If f is adjacent with an other 2-face, see Fig. 1 for illustration, then we remove all three uv edges, identify the vertices u, v , and proceed as in Case 2. First we color e1 , then e3 , and finally e2 .
Fig. 1. Two adjacent 2-faces in H.
Assume that there is no 2-face adjacent to f . Let H1 be the graph obtained from H by removing all vertices and edges inside the cycle formed by e2 and e3 ; and let H2 be the graph obtained from H by removing all vertices and edges outside the cycle formed by e1 and e3 , see Fig. 2 for illustration. It is easy to see that the graphs H1 and H2 are 3-edge-connected, have fewer vertices than H so each of them admits a suitable coloring. The outerface of H2 is formed by the edges e1 and e3 , so they receive different colors in any suitable Please cite this article as: J. Czap, S. Jendrol’ and J. Valiska, Edge-coloring of plane multigraphs with many colors on facial cycles, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.003.
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Fig. 2. The graphs H, H1 , and H2 .
coloring of H2 . If the edges e1 and e3 have different colors in a suitable coloring of H1 , then (by renaming the colors in H2 if necessary) the colorings of H1 and H2 induce a suitable coloring of H — the edge e2 receives the color from H2 . If the edges e1 and e3 have the same color in a suitable coloring of H1 , then first we recolor e3 with an admissible color (such a color always exists, since e3 is incident with a 2-face in H1 ; if it is necessary, then we recolor e2 ) and proceed as before. Consequently, H has no 2-faces. Now suppose that H contains parallel edges e1 = uv and e2 = uv which does not form a 2-face. Let C be the cycle formed by e1 and e2 . Let H1 be the graph obtained from H by removing all vertices and edges inside C and adding a new uv edge into C ; and let H2 be the graph consisting of the cycle C , all vertices and edges inside C , and a new edge joining u and v outside C , see Fig. 3 for illustration.
Fig. 3. The graphs H, H1 , and H2 .
The graphs H1 and H2 are 3-edge-connected, have fewer vertices than H so each of them admits a suitable coloring. Since the edge e1 is incident with a 2-face in Hi , there is a suitable coloring of Hi such that e1 and e2 receive different colors, i = 1, 2. These colorings of H1 and H2 (without the added uv edge) induce a suitable coloring of H. So H has no parallel edges. Using the same arguments as in the proof of Claim 4 we can show that H does not contain any face of size at least p + 1 (note that only in the proof of Claim 4 we need p + 2 colors). Since H is 3-edge-connected, the girth of its dual H ∗ is at least three, i.e. H ∗ is simple. Since each face of H has size at most p, in any facial p-coloring with respect to faces, the edges bounding every face are colored distinctly. Any such coloring of H induces a proper edge-coloring of H ∗ and vice versa. Vizing [17] proved that every simple graph with maximum degree p has a proper edge-coloring with at most p + 1 colors. Consequently, H has a suitable coloring, i.e. there is no counterexample to Theorem 4. □ Theorem 5.
If p ≥ 11 is an integer and G a 3-edge-connected plane graph, then arbrf (p, G) ≤ p + 1.
Proof. Suppose there is a counterexample to Theorem 5. Let H be a counterexample with minimum number of vertices, then minimum number of edges. Similarly as in the proof of Theorem 4 we can show that H has no parallel edges. Since H is 3-edge-connected its dual H ∗ has no 1-faces and 2-faces. The minimum degree of H ∗ is at least 3, because H has no loops and 2-faces. Consequently, H ∗ is a normal plane map, i.e. a plane graph with minimum degree at least 3 and minimum face size at least 3. Borodin [5] proved that every normal plane map has an edge uv such that the degree sum deg(u) + deg(v ) of its endvertices is at most 13. This implies that there are two adjacent faces in H such that the sum of their sizes is at most 13. Let f1 and f2 be such faces, and let e be the common edge. Consider the graph H /e. Clearly, H /e is 3-edge-connected and has no loops. H /e has fewer edges than H so it has a suitable coloring. Let fi′ be the face of H /e corresponding to fi in H, for i = 1, 2. Observe that at most 11 edges are incident with the faces f1′ and f2′ . If we color e with a color which does not appear on the faces f1′ and f2′ , then we obtain a suitable coloring of H, i.e. there is no counterexample to Theorem 5. Such a color exists since deg(f1′ ) + deg(f2′ ) ≤ 11 < p + 1. □ We believe that the following is true. Conjecture 1. If p ≥ 11 is an integer and G a 3-vertex-connected plane graph, then arbrf (p, G) ≤ p. Please cite this article as: J. Czap, S. Jendrol’ and J. Valiska, Edge-coloring of plane multigraphs with many colors on facial cycles, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.003.
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Acknowledgments This work was supported by the Slovak Research and Development Agency under the contract No. APVV-15-0116 and by the Slovak research grants VEGA 1/0368/16 and VEGA 1/0201/19. References [1] N. Alon, C. McDiarmid, B. Reed, Acyclic coloring of graphs, Random Struct. Algorithms 2 (1991) 277–288. [2] K. Appel, W. Haken, Every planar map is four colorable, Bull. Amer. Math. Soc. 82 (1976) 711–712. [3] T. Bartnicki, B. Bosek, S. Czerwiński, M. Farnik, J. Grytczuk, Z. Miechowicz, Generalized arboricity of graphs with large girth, Discrete Math. 342 (2019) 1343–1350. [4] J.A. Bondy, U.S.R. Murty, Graph Theory, Springer, 2008. [5] O.V. Borodin, On the total coloring of planar graphs, J. Reine Angew. Math. 394 (1989) 180–185. [6] J. Czap, S. Jendrol’, Facially-constrained colorings of plane graphs: a survey, Discrete Math. 340 (2017) 2691–2703. [7] J. Czap, S. Jendrol’, Facial colorings of plane graphs, J. Interconnect. Netw. 19 (2019) 1940003. [8] L. Ding, G. Wang, J. Wu, Generalized acyclic edge colorings via entropy compression, J. Comb. Optim. 35 (2018) 906–920. [9] I. Fabrici, S. Jendrol’, M. Voigt, Facial list colourings of plane graphs, Discrete Math. 339 (2016) 2826–2831. [10] S. Gerke, M. Raemy, Generalised acyclic edge colourings of graphs with large girth, Discrete Math. 307 (2007) 1668–1671. [11] I. Giotis, L. Kirousis, K.I. Psaromiligkos, D.M. Thilikos, On the algorithmic Lovász local lemma and acyclic edge coloring, in: Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics, 2015, pp. 16–25. [12] C. Greenhill, O. Pikhurko, Bounds on the generalised acyclic chromatic numbers of bounded degree graphs, Graphs Combin. 21 (2005) 407–419. [13] J.L. Gross, T.W. Tucker, Topological Graph Theory, Dover Publications, 2001. [14] C.St.J.A. Nash-Williams, Decomposition of finite graphs into forests, J. Lond. Math. Soc. 39 (1964) 12. [15] J. Nešetřil, P. Ossona de Mendez, X. Zhu, Colouring edges with many colours in cycles, J. Combin. Theory Ser. B 109 (2014) 102–119. [16] C.E. Shannon, A theorem on coloring the lines of a network, J. Math. Phys. 28 (1949) 148–151. [17] V.G. Vizing, On an estimate of the chromatic class of a p-graph, Diskret. Analiz 3 (1964) 25–30. [18] T. Wang, Y. Zhang, Further result on acyclic chromatic index of planar graphs, Discrete Appl. Math. 201 (2016) 228–247. [19] X. Zhang, G. Wang, Y. Yu, J. Li, G. Liu, On r-acyclic edge colorings of planar graphs, Discrete Appl. Math. 160 (2012) 2048–2053.
Please cite this article as: J. Czap, S. Jendrol’ and J. Valiska, Edge-coloring of plane multigraphs with many colors on facial cycles, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.003.
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Edge-coloring of plane multigraphs with many colors on facial cycles✩ ∗
Július Czap a , , Stanislav Jendrol’ b , Juraj Valiska b a
Department of Applied Mathematics and Business Informatics, Faculty of Economics, Technical University of Košice, Němcovej 32, 040 01 Košice, Slovakia b Institute of Mathematics, P. J. Šafárik University, Jesenná 5, 040 01 Košice, Slovakia
article
info
Article history: Received 13 May 2019 Received in revised form 24 October 2019 Accepted 4 November 2019 Available online xxxx Keywords: Plane graph Edge-coloring
a b s t r a c t For a fixed positive integer p, a coloring of the edges of a multigraph 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 coloring of G is the p-arboricity of G. From a result of Bartnicki et al. (2019) it follows that there are planar graphs with unbounded p-arboricity. In this note we relax the definition of p-arboricity for plane multigraphs in sense that the requirement is not for all cycles but only for facial cycles and show that the smallest number of colors needed for such a coloring is a constant (depending on p only). © 2019 Elsevier B.V. All rights reserved.
1. Introduction All graphs considered in this paper are loopless, parallel edges are allowed provided that it is not stated otherwise. We use standard graph theory terminology according to [4]. However, the most frequent notions of the paper are defined through it. A plane graph is a particular drawing of a planar graph in the Euclidean plane such that no edges intersect. Let G be a connected plane graph with vertex set V (G), edge set E(G), and face set F (G). The boundary of a face f is the boundary in the usual topological sense. It is the collection of all edges and vertices contained in the closure of f that can be organized into a closed walk in G traversing along a simple closed curve lying just inside the face f . This closed walk is unique up to the choice of initial vertex and direction, and is called the boundary walk of the face f (see [13], p. 101). The size of a face f , denoted by deg(f ), is the length of its boundary walk. A k-face is a face of size k. Two vertices (two edges) are adjacent if they are connected by an edge (have a common endvertex). A vertex and an edge are incident if the vertex is an endvertex of the edge. A vertex (or an edge) and a face are incident if the vertex (or the edge) lies on the boundary of the face. Two edges of a plane graph G are facially adjacent if they are consecutive on the boundary walk of a face of G. An edge-coloring of a graph G is an assignment of colors to the edges, one color to each edge. An edge-coloring c of a graph G is proper if for any two adjacent edges e1 and e2 of G, c(e1 ) ̸ = c(e2 ) holds. A facial edge-coloring c of a plane graph G is an edge-coloring such that for any two facially adjacent edges e1 and e2 of G, c(e1 ) ̸ = c(e2 ) holds. Observe that this coloring need not to be proper in a usual sense. We require only that facially ✩ An extended abstract of the research presented in this paper has been accepted at the European Conference on Combinatorics, Graph Theory and Applications (EUROCOMB 2019). ∗ Corresponding author. E-mail addresses: [email protected] (J. Czap), [email protected] (S. Jendrol’), [email protected] (J. Valiska). https://doi.org/10.1016/j.dam.2019.11.003 0166-218X/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: J. Czap, S. Jendrol’ and J. Valiska, Edge-coloring of plane multigraphs with many colors on facial cycles, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.003.
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adjacent edges must receive different colors. There are many results concerning different types of facial edge-colorings of plane graphs, see e.g. recent surveys [6] and [7]. A proper edge-coloring of a simple graph G is r-acyclic if every cycle C contained in G is colored with at least min{|C |, r } colors. The r-acyclic chromatic index a′r (G) of G is the minimum number of colors needed for an r-acyclic edge-coloring. Clearly, at least ∆ colors are required for any r-acyclic edge-coloring of a graph with maximum degree ∆. A 2-acyclic edgecoloring coincides with a proper edge-coloring, thus by the well-known Vizing’s theorem [17], ∆(G) ≤ a′2 (G) ≤ ∆(G) + 1 holds for every (simple) graph G. Alon, McDiarmid, and Reed [1] showed that the 3-acyclic chromatic index is linear in ∆. So far, the best upper bound for a′3 (G) is ⌈3.74(∆(G) − 1)⌉ + 1, which was obtained by Giotis et al. [11]. 3-acyclic edge-coloring has been deeply studied for planar graphs. The best known upper bound for this class of graphs is ∆ + 6, see [18]. For r ≥ 4 the situation is different, in this case the r-acyclic chromatic index may not be linear. In general, for every r r ≥ 4 there exist graphs with r-acyclic chromatic index at least cr · ∆⌊ 2 ⌋ , where cr is a constant (it depends only on r), see [12]. On the other hand, if the girth of a graph is sufficiently large, then its r-acyclic chromatic index is linear in ∆ if r is fixed. Gerke and Raemy [10] proved that for every graph G with girth at least 3(r − 1)∆(G) it holds that a′r (G) ≤ 6(r − 1)∆(G). In their theorem the girth is related to both r and ∆. Ding, Wang, and Wu [8] also proved a linear upper bound in terms of girth and maximum degree, but in their result the girth depends only on r. They proved that every graph with girth at least 2r − 1 admits an r-acyclic edge-coloring with at most (9r − 7)∆ + 10r − 12 colors. There is only one known result on r-acyclic edge-coloring of planar graphs with r ≥ 4. Zhang et al. [19] showed that a′4 (G) ≤ 37∆(G) for every planar graph G. The improper version of this problem is known as a generalized arboricity in the literature. The arboricity of a graph G is the minimum number of colors needed to color the edges of G so that every color class induces a forest. Therefore, if we require that every cycle gets at least two colors, then the minimum number of colors needed is the arboricity of G, and its determination is solved by the well-known Nash-Williams’ formula [14]. This concept was generalized by Nešetřil, Ossona de Mendez, and Zhu [15] in 2014. They introduced the p-arboricity of a graph G as the minimum number of colors needed to color the edges of G in such a way that every cycle C gets at least min{|C |, p + 1} colors. Obviously, the 1-arboricity is the classical arboricity. Bartnicki et al. [3] showed that the p-arboricity cannot be generally bounded by any constant (depending on p only) even for planar graphs, for all p ≥ 2. In this paper we relax the definition of p-arboricity for plane graphs in sense that the requirement is not for all cycles but only for facial cycles. We introduce the following parameter: For a fixed positive integer p, a facial edge-coloring of a plane graph G is called facial p-coloring with respect to faces, if there are at least min{|E(f )|, p} colors on the boundary of each face f , where E(f ) denotes the set of edges incident with the face f in G. The facial p-arboricity of a plane graph G with respect to faces, denoted by arbrf (p, G), is the smallest number of colors needed for a facial p-coloring with respect to faces. Our main result is that the facial p-arboricity of a connected plane graph G can be bounded by a constant depending on p only, which is a significant difference between the facial p-arboricity and the p-arboricity. 2. p-arboricity of planar graphs with large girth Bartnicki et al. [3] proved that there are planar graphs with unbounded p-arboricity, for all p ≥ 2. On the other hand, they proved that planar graphs with large girth have p-arboricity equal to p + 1. Theorem 1 ([3]). The p-arboricity of every planar graph of girth at least 2p+1 is p + 1, for all p ≥ 2. In the following we improve this result. In our theorem, the girth is bounded by a linear function of p. Theorem 2.
The p-arboricity of every planar graph of girth at least 5p + 1 is p + 1, for all p ≥ 2.
Proof. Suppose that there is a counterexample to Theorem 2. Let H be a counterexample with minimum number of vertices. The minimum degree of H is at least 2. To see this, suppose that H has a vertex v of degree 1. The graph H − v has fewer vertices than H so its p-arboricity is p + 1. The coloring of H − v can be extended to a coloring of H in such a way that we color the edge incident with v with an arbitrary color. Now we show that H has a path on p + 1 edges such that all of its internal vertices have degree 2 in H. If we iron the vertices of degree 2 in H, then we obtain a plane graph H ′ with minimum degree at least three. By Euler’s formula, |V (H ′ )| − |E(H ′ )| + |F (H ′ )| = 2. Using the Handshaking Lemma
∑
∑
deg(v ) = 2|E(H ′ )| =
v∈V (H ′ )
deg(f ),
f ∈F (H ′ )
we have
∑
(2 deg(v ) − 6) +
v∈V (H ′ )
∑
(deg(f ) − 6) = −12.
f ∈F (H ′ )
Please cite this article as: J. Czap, S. Jendrol’ and J. Valiska, Edge-coloring of plane multigraphs with many colors on facial cycles, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.003.
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Therefore, H ′ has a face f of size at most 5. Since the corresponding face in H has at least 5p + 1 edges, at least one edge of f is subdivided at least p times in H. So let P = v0 v1 . . . vp+1 be an induced path in H. By the minimality of the graph H, the p-arboricity of H − {v1 , . . . , vp } equals p + 1. Now color the edge vi vi+1 of P with color i + 1, for i = 0, . . . , p. Clearly, if a cycle of H contains the edges of P, then there are p + 1 colors on its edges. On the other hand, if a cycle C does not contain the edges of P, then C belongs to H − {v1 , . . . , vp }, consequently all p + 1 colors appear on C . □ 3. Facial p-arboricity of plane graphs with respect to faces Given a graph G and one of its edges e = uv , the contraction of e consists of replacing u and v by a new vertex adjacent to all the former neighbors of u and v , and removing the loop corresponding to the edge e. We denote the resulting graph by G/e. The dual G∗ of a plane graph G is obtained as follows: Corresponding to each face f of G there is a vertex f ∗ of G∗ , and corresponding to each edge e of G there is an edge e∗ of G∗ ; two vertices f ∗ and g ∗ are joined by the edge e∗ in G∗ if and only if their corresponding faces f and g are separated by the edge e in G (an edge separates the faces incident with it). The simplified medial M(G) of a plane graph G can be obtained as follows: Corresponding to each edge e of G there is a vertex m(e) of M(G); two vertices m(e1 ) and m(e2 ) are joined by an edge in M(G) if and only if their corresponding edges e1 and e2 are facially adjacent in G. It is known that every connected plane graph has a facial edge-coloring with at most 4 colors, moreover this bound is tight, see e.g. [9]. From the fact that in every facial edge-coloring every face has at least two colors on the boundary it follows that arbrf (2, G) ≤ 4 for every plane graph G. Theorem 3.
If p ≥ 3 is an integer and G a connected plane graph, then arbrf (p, G) ≤
3 p. 2
Moreover, this bound is tight.
Proof. Let p = 3. Let M(G) be the simplified medial of G. For every face f of M(G) of size at least four with boundary walk v1 v2 . . . vk we insert the edge v1 v3 into f . In such a way we obtain a new plane graph H. By the Four Color Theorem [2], H has a proper vertex-coloring with at most four colors. This vertex-coloring of H induces a facial edge-coloring of G. Since every 3-cycle in H uses three different colors, every face of G has at least three colors on the boundary. Now assume that p ≥ 4. Suppose there is a counterexample to Theorem 3. Let H be a counterexample with minimum number of vertices, then minimum number of edges. First we prove several structural properties of H. Claim 1.
H does not contain any cut-edge.
Proof. Suppose that H has a cut-edge e. Let fe be the face incident with e in H. By the minimality of H, the graph H /e has a facial edge-coloring with at most 23 p colors such that at least min{|E(f )|, p} colors appear on the boundary of each face f of H /e. Let fe′ be the face of H /e corresponding to fe in H. If less than p colors are used on the boundary of fe′ , then we color the edge e with a color which does not appear on fe′ . Otherwise we color e with a color that is not used on the edges facially adjacent to e. Such a color exists, since 32 p ≥ 6 and every edge is facially adjacent with at most four edges. □ Claim 2.
H does not contain any 2-face.
Proof. Suppose that H contains a 2-face f bounded with edges e1 = uv and e2 = uv . Let fe1 be the second face incident with e1 in H. The graph H −e1 has fewer edges than H, so it has a suitable coloring. Let fe′1 be the face of H −e1 corresponding to fe1 in H. If less than p + 1 colors are used on the boundary of fe′1 , then we color the edge e1 with a color which does not appear on fe′1 . Otherwise we color e1 with a color that is not used on the edges facially adjacent to e1 . □ Claim 3.
H does not contain parallel edges.
Proof. Suppose that H contains parallel edges e1 = uv and e2 = uv . By Claim 2, no two parallel edges form a 2-face, therefore there are at least one vertex inside and at least one vertex outside the cycle C formed by e1 and e2 . Let H1 be the graph obtained from H by deleting the vertices and edges outside C , and H2 be the graph obtained from H by deleting the vertices and edges inside C . The graphs H1 and H2 have fewer vertices than H, therefore each of them has a suitable coloring. Clearly, the edges incident with C are colored with two colors in both graphs H1 and H2 . Hence, by renaming the colors if necessary, the colorings of H1 and H2 induce a suitable coloring of H. □ Claim 4.
H does not contain any face of size at least p + 1.
Proof. Suppose that H contains a face f1 of size at least p + 1. Consider an edge e that f1 shares with a face f2 . By the minimality of H, the graph H /e has a facial edge-coloring with at most 23 p colors such that at least min{|E(f )|, p} colors appear on the boundary of each face f of H /e (the graph H /e has no loops by Claim 3). We extend the coloring of H /e to Please cite this article as: J. Czap, S. Jendrol’ and J. Valiska, Edge-coloring of plane multigraphs with many colors on facial cycles, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.003.
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a coloring of H in the following way. Let fi′ be the face of H /e corresponding to fi in H, for i = 1, 2. Observe that, there are at least p colors on f1′ , since its size is at least p. If at least p colors appear on f2′ , then we color e with a color that is not used on the edges facially adjacent to e. If less than p colors appear on f2′ , then we color e with a color that is not used on f2′ and on the edges facially adjacent to e. Such a color exists, since p + 2 ≤ 23 p for p ≥ 4. □ From Claims 1–4 it follows that the graph H is 2-edge-connected and each of its faces has size at most p. Consequently, its dual H ∗ is loopless and has maximum degree at most p. Since each face of H has size at most p, in any facial p-coloring with respect to faces, the edges bounding every face are colored distinctly. Such a coloring of H induces a proper edgecoloring of H ∗ and vice versa. Shannon [16] proved that every graph with maximum degree p has a proper edge-coloring with at most 32 p colors. Consequently, H has a suitable coloring, i.e. there is no counterexample to Theorem 3. p To see that the bound is tight, it suffices to paste together two p-cycles on 2 edges (if p is even). In this way we obtain a plane graph G which has three faces of size p. Thus, p different colors must appear on every face. Moreover, any two edges of G are incident with a common face, therefore no two edges are colored with the same color. The graph G has 23 p edges, hence arbrf (p, G) = 23 p. p−1 If p is odd, then it is sufficient to paste together a p-cycle and a (p − 1)-cycle on 2 edges. □ Since there are 2-edge-connected plane graphs G with arbrf (p, G) = 23 p, in the following we consider 3-edge-connected plane graphs and show that the bound 32 p can be significantly improved for this class of graphs. By Menger’s theorem, a graph is 3-edge-connected if and only if every pair of vertices is joined by at least 3 pairwise edge-disjoint paths. In the proofs of Theorems 4 and 5 we will use the fact that contracting an edge preserves edge-connectivity, see [4] p. 218. Proposition 1. If G is a k-edge-connected graph on at least three vertices, and e is any edge of G, then G/e is k-edge-connected. From Theorem 3 it follows that arbrf (3, G) ≤ 4. Therefore in the following we consider p ≥ 4. Theorem 4.
If p ≥ 4 is an integer and G a 3-edge-connected plane graph, then arbrf (p, G) ≤ p + 2.
Proof. Suppose there is a counterexample to Theorem 4. Let H be a counterexample with minimum number of vertices, then minimum number of edges. First we prove that H has no parallel edges. Suppose that H contains a 2-face f bounded with edges e1 = uv and e2 = uv . We distinguish three cases according to the number of uv edges in H. Case 1. First assume that there are at least four uv edges in H. In this case H − e1 is 3-edge-connected and has fewer edges than G, thus it has a suitable coloring. Let fe1 be the second face incident with e1 in H, and let fe′1 be the face of H − e1 corresponding to fe1 in H. We extend a suitable coloring of H − e1 to a suitable coloring of H as follows. If less than p + 1 colors are used on the boundary of fe′1 , then we color the edge e1 with a color which does not appear on fe′1 . Otherwise we color e1 with a color that is not used on the edges facially adjacent to e1 . Case 2. Now assume that there are two uv edges in H. Let H ′ be the graph obtained from H by removing the two uv edges and identifying the vertices u and v , i.e. H ′ = (H − e1 )/e2 . Clearly, H ′ is 3-edge-connected. Let fei ̸ = f be the face incident with ei in H, and let fe′i be the face of H ′ corresponding to fei in H, i = 1, 2. We extend a suitable coloring of H ′ in the following way. First we color the edge e1 . If less than p colors are used on the boundary of fe′1 , then we color the edge e1 with a color which does not appear on fe′1 . Otherwise we color e1 with a color that is not used on the edges facially adjacent to e1 . Now we color e2 . If less than p colors are used on the boundary of fe′2 , then we color the edge e2 with a color which does not appear on fe′2 and on the edge e1 . Otherwise we color e2 with a color that is not used on the edges facially adjacent to e2 . Case 3. Finally, assume that there are three uv edges e1 , e2 , e3 in H. If f is adjacent with an other 2-face, see Fig. 1 for illustration, then we remove all three uv edges, identify the vertices u, v , and proceed as in Case 2. First we color e1 , then e3 , and finally e2 .
Fig. 1. Two adjacent 2-faces in H.
Assume that there is no 2-face adjacent to f . Let H1 be the graph obtained from H by removing all vertices and edges inside the cycle formed by e2 and e3 ; and let H2 be the graph obtained from H by removing all vertices and edges outside the cycle formed by e1 and e3 , see Fig. 2 for illustration. It is easy to see that the graphs H1 and H2 are 3-edge-connected, have fewer vertices than H so each of them admits a suitable coloring. The outerface of H2 is formed by the edges e1 and e3 , so they receive different colors in any suitable Please cite this article as: J. Czap, S. Jendrol’ and J. Valiska, Edge-coloring of plane multigraphs with many colors on facial cycles, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.003.
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Fig. 2. The graphs H, H1 , and H2 .
coloring of H2 . If the edges e1 and e3 have different colors in a suitable coloring of H1 , then (by renaming the colors in H2 if necessary) the colorings of H1 and H2 induce a suitable coloring of H — the edge e2 receives the color from H2 . If the edges e1 and e3 have the same color in a suitable coloring of H1 , then first we recolor e3 with an admissible color (such a color always exists, since e3 is incident with a 2-face in H1 ; if it is necessary, then we recolor e2 ) and proceed as before. Consequently, H has no 2-faces. Now suppose that H contains parallel edges e1 = uv and e2 = uv which does not form a 2-face. Let C be the cycle formed by e1 and e2 . Let H1 be the graph obtained from H by removing all vertices and edges inside C and adding a new uv edge into C ; and let H2 be the graph consisting of the cycle C , all vertices and edges inside C , and a new edge joining u and v outside C , see Fig. 3 for illustration.
Fig. 3. The graphs H, H1 , and H2 .
The graphs H1 and H2 are 3-edge-connected, have fewer vertices than H so each of them admits a suitable coloring. Since the edge e1 is incident with a 2-face in Hi , there is a suitable coloring of Hi such that e1 and e2 receive different colors, i = 1, 2. These colorings of H1 and H2 (without the added uv edge) induce a suitable coloring of H. So H has no parallel edges. Using the same arguments as in the proof of Claim 4 we can show that H does not contain any face of size at least p + 1 (note that only in the proof of Claim 4 we need p + 2 colors). Since H is 3-edge-connected, the girth of its dual H ∗ is at least three, i.e. H ∗ is simple. Since each face of H has size at most p, in any facial p-coloring with respect to faces, the edges bounding every face are colored distinctly. Any such coloring of H induces a proper edge-coloring of H ∗ and vice versa. Vizing [17] proved that every simple graph with maximum degree p has a proper edge-coloring with at most p + 1 colors. Consequently, H has a suitable coloring, i.e. there is no counterexample to Theorem 4. □ Theorem 5.
If p ≥ 11 is an integer and G a 3-edge-connected plane graph, then arbrf (p, G) ≤ p + 1.
Proof. Suppose there is a counterexample to Theorem 5. Let H be a counterexample with minimum number of vertices, then minimum number of edges. Similarly as in the proof of Theorem 4 we can show that H has no parallel edges. Since H is 3-edge-connected its dual H ∗ has no 1-faces and 2-faces. The minimum degree of H ∗ is at least 3, because H has no loops and 2-faces. Consequently, H ∗ is a normal plane map, i.e. a plane graph with minimum degree at least 3 and minimum face size at least 3. Borodin [5] proved that every normal plane map has an edge uv such that the degree sum deg(u) + deg(v ) of its endvertices is at most 13. This implies that there are two adjacent faces in H such that the sum of their sizes is at most 13. Let f1 and f2 be such faces, and let e be the common edge. Consider the graph H /e. Clearly, H /e is 3-edge-connected and has no loops. H /e has fewer edges than H so it has a suitable coloring. Let fi′ be the face of H /e corresponding to fi in H, for i = 1, 2. Observe that at most 11 edges are incident with the faces f1′ and f2′ . If we color e with a color which does not appear on the faces f1′ and f2′ , then we obtain a suitable coloring of H, i.e. there is no counterexample to Theorem 5. Such a color exists since deg(f1′ ) + deg(f2′ ) ≤ 11 < p + 1. □ We believe that the following is true. Conjecture 1. If p ≥ 11 is an integer and G a 3-vertex-connected plane graph, then arbrf (p, G) ≤ p. Please cite this article as: J. Czap, S. Jendrol’ and J. Valiska, Edge-coloring of plane multigraphs with many colors on facial cycles, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.003.
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Acknowledgments This work was supported by the Slovak Research and Development Agency under the contract No. APVV-15-0116 and by the Slovak research grants VEGA 1/0368/16 and VEGA 1/0201/19. References [1] N. Alon, C. McDiarmid, B. Reed, Acyclic coloring of graphs, Random Struct. Algorithms 2 (1991) 277–288. [2] K. Appel, W. Haken, Every planar map is four colorable, Bull. Amer. Math. Soc. 82 (1976) 711–712. [3] T. Bartnicki, B. Bosek, S. Czerwiński, M. Farnik, J. Grytczuk, Z. Miechowicz, Generalized arboricity of graphs with large girth, Discrete Math. 342 (2019) 1343–1350. [4] J.A. Bondy, U.S.R. Murty, Graph Theory, Springer, 2008. [5] O.V. Borodin, On the total coloring of planar graphs, J. Reine Angew. Math. 394 (1989) 180–185. [6] J. Czap, S. Jendrol’, Facially-constrained colorings of plane graphs: a survey, Discrete Math. 340 (2017) 2691–2703. [7] J. Czap, S. Jendrol’, Facial colorings of plane graphs, J. Interconnect. Netw. 19 (2019) 1940003. [8] L. Ding, G. Wang, J. Wu, Generalized acyclic edge colorings via entropy compression, J. Comb. Optim. 35 (2018) 906–920. [9] I. Fabrici, S. Jendrol’, M. Voigt, Facial list colourings of plane graphs, Discrete Math. 339 (2016) 2826–2831. [10] S. Gerke, M. Raemy, Generalised acyclic edge colourings of graphs with large girth, Discrete Math. 307 (2007) 1668–1671. [11] I. Giotis, L. Kirousis, K.I. Psaromiligkos, D.M. Thilikos, On the algorithmic Lovász local lemma and acyclic edge coloring, in: Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics, 2015, pp. 16–25. [12] C. Greenhill, O. Pikhurko, Bounds on the generalised acyclic chromatic numbers of bounded degree graphs, Graphs Combin. 21 (2005) 407–419. [13] J.L. Gross, T.W. Tucker, Topological Graph Theory, Dover Publications, 2001. [14] C.St.J.A. Nash-Williams, Decomposition of finite graphs into forests, J. Lond. Math. Soc. 39 (1964) 12. [15] J. Nešetřil, P. Ossona de Mendez, X. Zhu, Colouring edges with many colours in cycles, J. Combin. Theory Ser. B 109 (2014) 102–119. [16] C.E. Shannon, A theorem on coloring the lines of a network, J. Math. Phys. 28 (1949) 148–151. [17] V.G. Vizing, On an estimate of the chromatic class of a p-graph, Diskret. Analiz 3 (1964) 25–30. [18] T. Wang, Y. Zhang, Further result on acyclic chromatic index of planar graphs, Discrete Appl. Math. 201 (2016) 228–247. [19] X. Zhang, G. Wang, Y. Yu, J. Li, G. Liu, On r-acyclic edge colorings of planar graphs, Discrete Appl. Math. 160 (2012) 2048–2053.
Please cite this article as: J. Czap, S. Jendrol’ and J. Valiska, Edge-coloring of plane multigraphs with many colors on facial cycles, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.11.003.