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

LaBRI, Université Bordeaux I, Talence, France
Discrete Applied Mathematics (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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    • "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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