A Uniquely $3$-List Colorable, Planar and $K_4$-Free Graph

Abstract

Let $G$ be a graph with $v$ vertices. If there exists a collection of lists of colors $\{S_1,S_2,\dots ,S_v\}$ on its vertices, each of size $k$, such that there exists a unique proper coloring for $G$ from this list of colors, then $G$ is called a $uniquelyk-listcolorablegraph$. In this note, we present a uniquely $3$-list colorable, planar, and $K_4$-free graph. It is a counterexample to a conjecture by Ch. Eslahchi, M. Ghebleh, and H. Hajiabolhassan [3].