Elsevier

Discrete Mathematics

Volume 309, Issue 20, 28 October 2009, Pages 6044-6047
Discrete Mathematics

Adaptable choosability of planar graphs with sparse short cycles

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Abstract

Given a (possibly improper) edge colouring F of a graph G, a vertex colouring of G is adapted to F if no colour appears at the same time on an edge and on its two endpoints. A graph G is called adaptablyk-choosable (for some positive integer k) if for any list assignment L to the vertices of G, with |L(v)|k for all v, and any edge colouring F of G, G admits a colouring c adapted to F where c(v)L(v) for all v. This paper proves that a planar graph G is adaptably 3-choosable if any two triangles in G have distance at least 2 and no triangle is adjacent to a 4-cycle.

Keywords

Adapted colouring
List colouring
Planar graphs

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