Article
Planar graphs without 4, 5 and 8cycles are acyclically 4choosable
LaBRI, Université Bordeaux I, Talence, France
Discrete Applied Mathematics
(Impact Factor: 0.68).
08/2009;
161(78):659667.
DOI: 10.1016/j.endm.2009.07.111
Source: DBLP

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. 
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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 Lcolorable if there exists a proper acyclic coloring π of G such that π(v) ∈ L(v) for all v ∈ V. If G is acyclically Lcolorable for any list assignment L with L(v) ⩾ k for all v ∈ V (G), then G is acyclically kchoosable. In this paper, we prove that every planar graph G is acyclically 6choosable if G does not contain 4cycles adjacent to icycles for each i ∈ {3, 4, 5, 6}. This improves the result by Wang and Chen (2009).Science China Mathematics 12/2013; 57(1). DOI:10.1007/s1142501345726 · 0.71 Impact Factor 
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ABSTRACT: Every planar graph is known to be acyclically 7choosable and is conjectured to be acyclically 5choosable (O. V. Borodin, D. G. FonDerFlaass, 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 5color theorem and Thomassen's (J Combin Theory Ser B 62 (1994), 180–181) 5choosability 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 3choosable. In particular, the acyclic 4choosability was proved for the following planar graphs: without 3, 4, and 5cycles (M. Montassier, P. Ochem, and A. Raspaud, J Graph Theory 51 (2006), 281–300), without 4, 5, and 6cycles, or without 4, 5, and 7cycles, or without 4, 5, and intersecting 3cycles (M. Montassier, A. Raspaud, W. Wang, Topics Discrete Math (2006), 473–491), and neither 4 and 5cycles nor 8cycles 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 5cycles is acyclically 4choosable.Journal of Graph Theory 04/2013; 72(4). DOI:10.1002/jgt.21647 · 0.67 Impact Factor
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