A Sufficient Condition for Planar Graphs

• Authors: Min Chen;André Raspaud
• Venue: Journal of Graph Theory
• Year: 2012

A proper vertex coloring of a graph G = (V, E) is acyclic if G contains no bicolored cycle. Given a list assignment L = {L(v)|v∈V} of G, we say G is acyclically L-list 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-list colorable for any list assignment with |L(v)|≥k for all v∈V, then G is acyclically k-choosable. In this article we prove that every planar graph without 4-cycles and without intersecting triangles is acyclically 5-choosable. This improves the result in [M. Chen and W. Wang, Discrete Math 308 (2008), 6216–6225], which says that every planar graph without 4-cycles and without two triangles at distance less than 3 is acyclically 5-choosable. © 2011 Wiley Periodicals, Inc. J Graph Theory (Contract grant sponsors: French Embassy in Beijing (to M. C.); The University of Bordeaux I (to M. C.); The LaBRI (to M. C.); CROUS (to M. C.). © 2012 Wiley Periodicals, Inc.)