Planar graphs without 4, 5 and 8-cycles are acyclically 4-choosable

LaBRI, Université Bordeaux I, Talence, France
Discrete Applied Mathematics (Impact Factor: 0.8). 08/2009; 161(7-8):659-667. DOI: 10.1016/j.endm.2009.07.111
Source: DBLP


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(nu) : nu is an element of V}, there exists a proper acyclic coloring pi of G such that pi(nu) is an element of L(nu) for all nu is an element of V. If G is acyclically L-list colorable for any list assignment with vertical bar L(nu)vertical bar >= k for all nu is an element of V, then G is acyclically k-choosable. In 1976, Steinberg Jensen and Toft (1995) [20] conjectured that every planar graph without 4- and 5-cycles is 3-colorable. This conjecture cannot be improved to 3-choosability basing on the examples given by Voigt (2007) [30] and, independently, by Montassier (2005) [24]. In this paper, we prove that planar graphs without 4- and 5-cycles are acyclically 4-choosable. This result (obtained independently by Borodin and Ivanova (2012) [9]) is also a new approach to the conjecture proposed by Montassier et al. (2006) in [27], which says that every planar graph without 4-cycles is acyclically 4-choosable.

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    • "Some sufficient conditions have been obtained for a planar graph to be acyclic 4-colorable. In 1999, Borodin, Kostochka, and Woodall [4] showed that planar graphs under the absence of 3-and 4-cycles are acyclic 4-colorable; In 2006, Montassier, Raspaud, and Wang [15] proved that planar graphs, without 4-,5-, and 6-cycles, or without 4-, 5-, and 7-cycles, or without 4-, 5-, and intersecting 3-cycles, are acyclic 4-colorable; In 2009, Chen and Raspaud [9] proved that if a planar graph G has no 4-, 5-, and 8-cycles, then G is acyclic 4-colorable; Also in 2009, Borodin[5] showed that planar graphs without 4-and 6-cycles are acyclic 4-colorable; Additionally, Borodin in 2011[6] and 2013[7] proved that planar graphs without 4-and 5-cycles are acyclic 4-colorable and acyclically 4-choosable, respectively. "
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    ABSTRACT: A tree-coloring of a maximal planar graph is a proper vertex $4$-coloring such that every bichromatic subgraph, induced by this coloring, is a tree. A maximal planar graph $G$ is tree-colorable if $G$ has a tree-coloring. In this article, we prove that a tree-colorable maximal planar graph $G$ with $\delta(G)\geq 4$ contains at least four odd-vertices. Moreover, for a tree-colorable maximal planar graph of minimum degree 4 that contains exactly four odd-vertices, we show that the subgraph induced by its four odd-vertices is not a claw and contains no triangles.
    Full-text · Article · Mar 2014
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    ABSTRACT: The acyclic 4-choosability was proved, in particular, for the following planar graphs: without 3- and 4-cycles (Montassier et al., 2006 [29]), without 4-, 5-, and 6-cycles (Montassier et al., 2006 [29]), either without 4-, 6-, and 7-cycles, or without 4-, 6-, and 8-cycles (Chen, Raspaud, and Wang, 2009), and with neither 4-cycles nor 6-cycles adjacent to a triangle (Borodin et al., 2010[13]). There exist planar acyclically non-4-colorable bipartite graphs (Kostochka and Mel'nikov, 1976[25]). This partly explains the fact that in all previously known sufficient conditions for the acyclic 4-choosability of planar graphs the 4-cycles are completely forbidden. In this paper we allow 4-cycles nonadjacent to relatively short cycles; namely, it is proved that a planar graph is acyclically 4-choosable if it does not contain an i-cycle adjacent to a j-cycle, where 3 <= j <= 6 if i = 3 and 4 <= j <= 7 if i = 4. In particular, this absorbs all the above-mentioned results.
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    ABSTRACT: After a brief historical account, a few simple structural theorems about plane graphs useful for coloring are stated, and two simple applications of discharging are given. Afterwards, the following types of proper colorings of plane graphs are discussed, both in their classical and choosability (list coloring) versions: simultaneous colorings of vertices, edges, and faces (in all possible combinations, including total coloring), edge-coloring, cyclic coloring (all vertices in any small face have different colors), 3-coloring, acyclic coloring (no 2-colored cycles), oriented coloring (homomorphism of directed graphs to small tournaments), a special case of circular coloring (the colors are points of a small cycle, and the colors of any two adjacent vertices must be nearly opposite on this cycle), 2-distance coloring (no 2-colored paths on three vertices), and star coloring (no 2-colored paths on four vertices). The only improper coloring discussed is injective coloring (any two vertices having a common neighbor should have distinct colors).
    No preview · Article · Feb 2013 · Discrete Mathematics
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