On [formula omitted]-choosability of planar graphs without adjacent short cycles

Abstract

A list assignment of a graph G is a function L that assigns a list L(v) of colors to each vertex v∈V(G). An (L,d)∗-coloring is a mapping π that assigns a color π(v)∈L(v) to each vertex v∈V(G) so that at most d neighbors of v receive the color π(v). A graph G is said to be (k,d)∗-choosable if it admits an (L,d)∗-coloring for every list assignment L with |L(v)|≥k for all v∈V(G). In 2001, Lih et al. (2001) [6] proved that planar graphs without 4- and l-cycles are (3,1)∗-choosable, where l∈{5,6,7}. Later, Dong and Xu (2009) [3] proved that planar graphs without 4- and l-cycles are (3,1)∗-choosable, where l∈{8,9}.There exist planar graphs containing 4-cycles that are not (3,1)∗-choosable (Cowen et al., 1986 [1]). This partly explains the fact that in all above known sufficient conditions for the (3,1)∗-choosability of planar graphs the 4-cycles are completely forbidden. In this paper we allow 4-cycles nonadjacent to relatively short cycles. More precisely, we prove that every planar graph without 4-cycles adjacent to 3- and 4-cycles is (3,1)∗-choosable. This is a common strengthening of all above mentioned results. Moreover as a consequence we give a partial answer to a question of Xu and Zhang (2007) [11] and show that every planar graph without 4-cycles is (3,1)∗-choosable.