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- W2084747564 abstract "An acyclic coloring of a graph G is a coloring of its vertices such that: (i) no two adjacent vertices in G receive the same color and (ii) no bicolored cycles exist in G . A list assignment of G is a function L that assigns to each vertex v ∈ V ( G ) a list L ( v ) of available colors. Let G be a graph and L be a list assignment of G . The graph G is acyclically L -list colorable if there exists an acyclic coloring ϕ of G such that ϕ ( v ) ∈ L ( v ) for all v ∈ V ( G ) . If G is acyclically L -list colorable for any list assignment L with | L ( v ) | ≥ k for all v ∈ V ( G ) , then G is said to be acyclically k -choosable. Borodin et al. proved that every planar graph with girth at least 7 is acyclically 3-choosable (Borodin et al., submitted for publication [4] ). More recently, Borodin and Ivanova showed that every planar graph without cycles of length 4 to 11 is acyclically 3-choosable (Borodin and Ivanova, submitted for publication [7] ). In this note, we connect these two results by a sequence of intermediate sufficient conditions that involve the minimum distance between 3-cycles: we prove that every planar graph with neither cycles of lengths 4 to 7 (resp. to 8, to 9, to 10) nor triangles at distance less than 7 (resp. 5, 3, 2) is acyclically 3-choosable." @default.
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- W2084747564 date "2010-05-01" @default.
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- W2084747564 title "A note on the acyclic 3-choosability of some planar graphs" @default.
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- W2084747564 doi "https://doi.org/10.1016/j.dam.2010.02.005" @default.
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