On -choosability of planar graphs without adjacent short cycles (original) (raw)

2014, Discrete Applied Mathematics

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.