Acyclic edge coloring of planar graphs with colors

Discrete Mathematics, 2014

A strong edge coloring of a graph is a proper edge coloring where the edges at distance at most two receive distinct colors. It is known that every planar graph with maximum degree ∆ has a strong edge coloring with at most 4 ∆ + 4 colors. We show that 3 ∆ + 6 colors suffice if the graph has girth 6, and 3 ∆ colors suffice if the girth is at least 7. Moreover, we show that cubic planar graphs with girth at least 6 can be strongly edge colored with at most 9 colors.

Discrete Applied Mathematics, 2019

In this paper, we study the achromatic and the pseudoachromatic numbers of planar and outerplanar graphs as well as planar graphs of girth 4 and graphs embedded on a surface. We give asymptotically tight results and lower bounds for maximal embedded graphs.

Discrete Mathematics, 2015

A graph is (d1,. .. , d k)-colorable if its vertex set can be partitioned into k nonempty subsets so that the subgraph induced by the ith part has maximum degree at most di for each i ∈ {1,. .. , k}. It is known that for each pair (d1, d2), there exists a planar graph with girth 4 that is not (d1, d2)-colorable. This sparked the interest in finding the pairs (d1, d2) such that planar graphs with girth at least 5 are (d1, d2)-colorable. Given d1 ≤ d2, it is known that planar graphs with girth at least 5 are (d1, d2)-colorable if either d1 ≥ 2 and d1 + d2 ≥ 8 or d1 = 1 and d2 ≥ 10. We improve an aforementioned result by providing the first pair (d1, d2) in the literature satisfying d1 + d2 ≤ 7 where planar graphs with girth at least 5 are (d1, d2)-colorable. Namely, we prove that planar graphs with girth at least 5 are (3, 4)-colorable.

Journal of Combinatorics, 2017

An incidence of an undirected graph G is a pair (v, e) where v is a vertex of G and e an edge of G incident with v. Two incidences (v, e) and (w, f) are adjacent if one of the following holds: (i) v = w, (ii) e = f or (iii) vw = e or f. An incidence coloring of G assigns a color to each incidence of G in such a way that adjacent incidences get distinct colors. In 2012, Yang [15] proved that every planar graph has an incidence coloring with at most ∆ + 5 colors, where ∆ denotes the maximum degree of the graph. In this paper, we show that ∆ + 4 colors suffice if the graph is planar and without a C 3 adjacent to a C 4. Moreover, we prove that every planar without C 4 and C 5 and maximum degree at least 5 admits an incidence coloring with at most ∆ + 3 colors.

arXiv preprint arXiv: …, 2012

An additive coloring of a graph G is an assignment of positive integers {1, 2, . . . , k} to the vertices of G such that for every two adjacent vertices the sums of numbers assigned to their neighbors are different. The minimum number k for which there exists an additive coloring of G is denoted by η(G). We prove that η(G) 468 for every planar graph G. This improves a previous bound η(G) 5544 due to Norin. The proof uses Combinatorial Nullstellensatz and coloring number of planar hypergrahs. We also demonstrate that η(G) 36 for 3-colorable planar graphs, and η(G) 4 for every planar graph of girth at least 13. In a group theoretic version of the problem we show that for each r 2 there is an r-chromatic graph G r with no additive coloring by elements of any Abelian group of order r.

Discrete Mathematics, 2018

A strong edge-coloring of a graph is a proper edge-coloring such that edges at distance at most 2 receive different colors. It is known that every planar graph has a strong edgecoloring by using at most 4∆ + 4 colors, where ∆ denotes the maximum degree of the graph. In this paper, we will show that 19 colors are enough to color a planar graph with maximum degree 4.

Discrete Mathematics, 2010

We prove that every planar graph in which no i-cycle is adjacent to a j-cycle whenever 3 ≤ i ≤ j ≤ 7 is 3-colorable and pose some related problems on the 3-colorability of planar graphs. 1 Introduction In 1976, Appel and Haken proved that every planar graph is 4-colorable [3, 4], and as early as 1959, Grötzsch [15] proved that every planar graph without 3-cycles is 3-colorable. As proved by Garey, Johnson and Stockmeyer [14], the problem of deciding whether a planar graph is 3-colorable is NP-complete. Therefore, some sufficient conditions for planar graphs to be 3-colorable were stated. In 1976, Steinberg [19] raised the following: Steinberg's Conjecture '76 Every planar graph without 4-and 5-cycles is 3-colorable. In 1969, Havel [16] posed the following problem: Havel's Problem '69 Does there exist a constant C such that every planar graph with the minimum distance between triangles at least C is 3-colorable?

Discrete Mathematics, 2009

In this paper, we mainly prove that planar graphs without 4-, 7-and 9-cycles are 3colorable.

Wang and Lih conjectured that for every g ≥ 5, there exists a number M (g) such that the chromatic number of the square of every planar graph of girth at least g and maximum degree ∆ ≥ M (g) is ∆ + 1. We disprove the conjecture for g ∈ {5, 6} and prove the existence of the number M (g) for g ≥ 7. More generally, we show that every planar graph of girth at least 7 and maximum degree ∆ ≥ 190 + 2 p/q has an L(p, q)-labeling of span at most 2p + q∆ -2. For q = 1, the bound is tight for all pairs of ∆ and p. We also show that the square of every planar graph of girth at least six and sufficiently large maximum degree ∆ is (∆ + 2)-colorable.

Discrete Mathematics, 2010

Let G = (V , E) be a graph. A proper vertex coloring of G is acyclic if G contains no bicolored cycle. Namely, every cycle of G must be colored with at least three colors. G is acyclically L-list colorable if for a given list assignment L = {L(v) : v ∈ V }, there exists a proper acyclic coloring π of G such that π (v) ∈ L(v) for all v ∈ V. If G is acyclically L-list colorable for any list assignment with |L(v)| ≥ k for all v ∈ V , then G is acyclically k-choosable. In this paper, we prove that planar graphs with neither {4, 5}-cycles nor 8-cycles having a triangular chord are acyclically 4-choosable. This implies that planar graphs either without {4, 5, 7}-cycles or without {4, 5, 8}-cycles are acyclically 4-choosable.