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

LaBRI, Université Bordeaux I, Talence, France
(Impact Factor: 0.68). 08/2009; 161(7-8):659-667. DOI: 10.1016/j.endm.2009.07.111
Source: DBLP

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 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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##### Article: Tree-colorable maximal planar graphs
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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.
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ABSTRACT: A proper vertex coloring of a graph G is acyclic if G contains no bicolored cycles. Given a list assignment L = {L(v) | v ∈ V} of G, we say that G is acyclically L-colorable if there exists a proper acyclic coloring π of G such that π(v) ∈ L(v) for all v ∈ V. If G is acyclically L-colorable for any list assignment L with |L(v)| ⩾ k for all v ∈ V (G), then G is acyclically k-choosable. In this paper, we prove that every planar graph G is acyclically 6-choosable if G does not contain 4-cycles adjacent to i-cycles for each i ∈ {3, 4, 5, 6}. This improves the result by Wang and Chen (2009).
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• ##### Article: Acyclic 4-Choosability of Planar Graphs with No 4- and 5-Cycles
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ABSTRACT: Every planar graph is known to be acyclically 7-choosable and is conjectured to be acyclically 5-choosable (O. V. Borodin, D. G. Fon-Der-Flaass, A. V. Kostochka, E. Sopena, J Graph Theory 40 (2002), 83–90). This conjecture if proved would imply both Borodin's (Discrete Math 25 (1979), 211–236) acyclic 5-color theorem and Thomassen's (J Combin Theory Ser B 62 (1994), 180–181) 5-choosability theorem. However, as yet it has been verified only for several restricted classes of graphs. Some sufficient conditions are also obtained for a planar graph to be acyclically 4- and 3-choosable. In particular, the acyclic 4-choosability was proved for the following planar graphs: without 3-, 4-, and 5-cycles (M. Montassier, P. Ochem, and A. Raspaud, J Graph Theory 51 (2006), 281–300), without 4-, 5-, and 6-cycles, or without 4-, 5-, and 7-cycles, or without 4-, 5-, and intersecting 3-cycles (M. Montassier, A. Raspaud, W. Wang, Topics Discrete Math (2006), 473–491), and neither 4- and 5-cycles nor 8-cycles having a triangular chord (M. Chen and A. Raspaud, Discrete Math. 310(15–16) (2010), 2113–2118). The purpose of this paper is to strengthen these results by proving that each planar graph without 4- and 5-cycles is acyclically 4-choosable.
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