Disproof of a Conjecture on List Coloring

Abstract

Let $G$ be the graph obtained from $K_{3,3}$ by deleting an edge. We find a list assignment with $|L(v)|=2$ for each vertex $v$ of $G$, such that $G$ is uniquely $L$-colorable, and show that for any list assignment $L^{\prime }$ of $G$, if $|Z^{\prime }(v)|\ge 2$ for all $v\in V(G)$ and there exists a vertex $v_0$ with $|L^{\prime }(v_0)|>2$, then $G$ is not uniquely $L^{\prime }$-colorable. However, $G$ is not $2$-choosable. This disproves a conjecture of Akbari, Mirrokni, and Sadjad (Problem $404$ in Discrete Math. $266(2003)441-451)$.