(3, 1)*-choosability of planar graphs without adjacent short cycles

Abstract

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 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. [Lih, K., Z. Song, W. Wang, and K. Zhang, A note on list improper coloring planar graphs, Appl. Math. Lett. 14 (2001), 269–273] proved that planar graphs without 4- and l-cycles are ( 3 , 1 ) ⁎ -choosable, where l ∈ { 5 , 6 , 7 } . Later, Dong and Xu [Dong, W., and B. Xu, A note on list improper coloring of plane graphs, Discrete Appl. Math. 157 (2009), 433–436] 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 (Crown, Crown and Woodall, 1986 [Cowen, L., R. Cowen, and D. Woodall, Defective colorings of graphs in surfaces: partitions into subgraphs of bounded valency, J. Graph Theory 10 (1986), 187–195]). 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 [Xu, B., and H. Zhang, Every toroidal graph without adjacent triangles is ( 4 , 1 ) ⁎ -choosable, Discrete Appl. Math. 155 (2007), 74–78] and show that every planar graph without 4-cycles is ( 3 , 1 ) ⁎ -choosable.