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

A list assignment of a graph $G$ is a function $L$ that assigns a list $L(v)$ of colors to each vertex $v\in V(G)$. An $(L,d)^*$-coloring is a mapping $π$ that assigns a color $π(v)\in L(v)$ to each vertex $v\in V(G)$ 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 $|L(v)|\ge k$ for all $v\in V(G)$. In 2001, Lih et al. \cite{LSWZ-01} proved that planar graphs without 4- and $l$-cycles are $(3,1)^*$-choosable, where $l\in \{5,6,7\}$. Later, Dong and Xu \cite{DX-09} proved that planar graphs without 4- and l-cycles are $(3,1)^*$-choosable, where $l\in \{8,9\}$. There exist planar graphs containing 4-cycles that are not $(3,1)^*$-choosable (Crown, Crown and Woodall, 1986 \cite{CCW-86}). 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 \cite{XZ-07} and show that every planar graph without 4-cycles is $(3,1)^*$-choosable.

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