List-Distinguishing Colorings of Graphs

Abstract

A coloring of the vertices of a graph $G$ is said to be distinguishing provided that no nontrivial automorphism of $G$ preserves all of the vertex colors. The distinguishing number of $G$, denoted $D(G)$, is the minimum number of colors in a distinguishing coloring of $G$. The distinguishing number, first introduced by Albertson and Collins in 1996, has been widely studied and a number of interesting results exist throughout the literature. In this paper, the notion of distinguishing colorings is extended to that of list-distinguishing colorings. Given a family $L=\{L(v)\}_{v\in V(G)}$ of lists assigning available colors to the vertices of $G$, we say that $G$ is $L$-distinguishable if there is a distinguishing coloring $f$ of $G$ such that $f(v)\in L(v)$ for all $v$. The list-distinguishing number of $G$, $D_{\ell}(G)$, is the minimum integer $k$ such that $G$ is $L$-distinguishable for any assignment $L$ of lists with $|L(v)|=k$ for all $v$. Here, we determine the list-distinguishing number for several families of graphs and highlight a number of distinctions between the problems of distinguishing and list-distinguishing a graph.