A Uniquely \(3\)-List Colorable, Planar and \(K_{4}\)-Free Graph

Let \( G \) be a graph with \( v \) vertices. If there exists a collection of lists of colors \(\{S_1, S_2, \ldots, S_v\}\) 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 \emph{uniquely \( k \)-list colorable graph}. 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].