Graphs \(K_{1*4,5}\), \(K_{1*5,4}\), \(K_{1*4,4}\), \(K_{2,3,4}\) have the Property \(M(3)\)

Abstract

Let \(G\) be a graph with \(n\) vertices and suppose that for each vertex \(v\) in \(G\), there exists a list of \(k\) colors, \(L(v)\), such that there is a unique proper coloring for \(G\) from this collection of lists, then \(G\) is called a uniquely \(k\)-list colorable graph. We say that a graph \(G\) has the property \(M(k)\) if and only if it is not uniquely \(k\)-list colorable. M. Ghebleh and E. S. Mahmoodian characterized uniquely \(3\)-list colorable complete multipartite graphs except for the graphs \(K_{1*4,5}\), \(K_{1*5,4}, K_{1*4,4}\), \(K_{2,3,4}\), and \(K_{2,2,r}\), \(4 \leq r \leq 8\). In this paper, we prove that the graphs \(K_{1*4,5}\), \(K_{1*5,4}\), \(K_{1*4,4}\), and \(K_{2,3,4}\) have the property \(M(3)\).