Planar graphs without 4, 5 and 8cycles are acyclically 4choosable
ABSTRACT 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 Llist 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 Llist 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 kchoosable. In 1976, Steinberg Jensen and Toft (1995) [20] conjectured that every planar graph without 4 and 5cycles is 3colorable. This conjecture cannot be improved to 3choosability 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 5cycles are acyclically 4choosable. 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 4cycles is acyclically 4choosable.

 "Some sufficient conditions have been obtained for a planar graph to be acyclic 4colorable. In 1999, Borodin, Kostochka, and Woodall [4] showed that planar graphs under the absence of 3and 4cycles are acyclic 4colorable; In 2006, Montassier, Raspaud, and Wang [15] proved that planar graphs, without 4,5, and 6cycles, or without 4, 5, and 7cycles, or without 4, 5, and intersecting 3cycles, are acyclic 4colorable; In 2009, Chen and Raspaud [9] proved that if a planar graph G has no 4, 5, and 8cycles, then G is acyclic 4colorable; Also in 2009, Borodin[5] showed that planar graphs without 4and 6cycles are acyclic 4colorable; Additionally, Borodin in 2011[6] and 2013[7] proved that planar graphs without 4and 5cycles are acyclic 4colorable and acyclically 4choosable, respectively. "
Article: Treecolorable maximal planar graphs
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ABSTRACT: A treecoloring 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 treecolorable if $G$ has a treecoloring. In this article, we prove that a treecolorable maximal planar graph $G$ with $\delta(G)\geq 4$ contains at least four oddvertices. Moreover, for a treecolorable maximal planar graph of minimum degree 4 that contains exactly four oddvertices, we show that the subgraph induced by its four oddvertices is not a claw and contains no triangles.  [Show abstract] [Hide abstract]
ABSTRACT: The acyclic 4choosability was proved, in particular, for the following planar graphs: without 3 and 4cycles (Montassier et al., 2006 [29]), without 4, 5, and 6cycles (Montassier et al., 2006 [29]), either without 4, 6, and 7cycles, or without 4, 6, and 8cycles (Chen, Raspaud, and Wang, 2009), and with neither 4cycles nor 6cycles adjacent to a triangle (Borodin et al., 2010[13]). There exist planar acyclically non4colorable bipartite graphs (Kostochka and Mel'nikov, 1976[25]). This partly explains the fact that in all previously known sufficient conditions for the acyclic 4choosability of planar graphs the 4cycles are completely forbidden. In this paper we allow 4cycles nonadjacent to relatively short cycles; namely, it is proved that a planar graph is acyclically 4choosable if it does not contain an icycle adjacent to a jcycle, where 3 <= j <= 6 if i = 3 and 4 <= j <= 7 if i = 4. In particular, this absorbs all the abovementioned results.Journal of Graph Theory 11/2012; 68(2):169  176. DOI:10.1002/jgt.20549 · 0.67 Impact Factor 
Article: Colorings of plane graphs: A survey
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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), edgecoloring, cyclic coloring (all vertices in any small face have different colors), 3coloring, acyclic coloring (no 2colored 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), 2distance coloring (no 2colored paths on three vertices), and star coloring (no 2colored paths on four vertices). The only improper coloring discussed is injective coloring (any two vertices having a common neighbor should have distinct colors).Discrete Mathematics 02/2013; 313(4):517–539. DOI:10.1016/j.disc.2012.11.011 · 0.57 Impact Factor