On a Conjecture About \((k, t)\)-Choosability

Abstract

A \((k,t)\)-list assignment \(L\) of a graph \(G\) is a list of \(k\) colors available at each vertex \(v\) in \(G\) such that \(|\bigcup_{v\in V(G)}L(v)| = t\). A proper coloring \(c\) such that \(c(v) \in L(v)\) for each \(v \in V(G)\) is said to be an \(L\)-coloring. We say that a graph \(G\) is \(L\)-colorable if \(G\) has an \(L\)-coloring. A graph \(G\) is \((k,t)\)-choosable if \(G\) is \(L\)-colorable for every \((k,t)\)-list assignment \(L\).
Let \(G\) be a graph with \(n\) vertices and \(G\) does not contain \(C_5\) or \(K_{k-2}\) and \(K_{k+1}\). We prove that \(G\) is \((k, kn – k^2 – 2k)\)-choosable for \(k \geq 3\).\(G\) is not \((k, kn – k^2 – 2k)\)-choosable for \(k = 2\).This result solves a conjecture posed by Chareonpanitseri, Punnim, and Uiyyasathian [W. Chareonpan-itseri, N. Punnim, C. Uiyyasathian, On \((k,t)\)-choosability of Graphs: Ars Combinatoria., \(99, (2011) 321-333]\).