On ${(3,1)}^{\ast }$-choosability of planar graphs without adjacent short cycles

Abstract

A list assignment of a graph $G$ is a function $L$ that assigns a list $L\left(v\right)$ of colors to each vertex $v\in V\left(G\right)$. An ${(L,d)}^{\ast }$-coloring is a mapping $\pi$ that assigns a color $\pi \left(v\right)\in L\left(v\right)$ to each vertex $v\in V\left(G\right)$ so that at most $d$ neighbors of $v$ receive the color $\pi \left(v\right)$. A graph $G$ is said to be ${(k,d)}^{\ast }$-choosable if it admits an ${(L,d)}^{\ast }$-coloring for every list assignment $L$ with $\left|L\left(v\right)\right|\ge k$ for all $v\in V\left(G\right)$. In 2001, Lih et al. (2001)  [6] proved that planar graphs without 4- and $l$-cycles are ${(3,1)}^{\ast }$-choosable, where $l\in \{5,6,7\}$. Later, Dong and Xu (2009)  [3] proved that planar graphs without 4- and $l$-cycles are ${(3,1)}^{\ast }$-choosable, where $l\in \{8,9\}$.

There exist planar graphs containing 4-cycles that are not ${(3,1)}^{\ast }$-choosable (Cowen et al., 1986  [1]). This partly explains the fact that in all above known sufficient conditions for the ${(3,1)}^{\ast }$-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 ${(3,1)}^{\ast }$-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 ${(3,1)}^{\ast }$-choosable.