List Coloring and $n$-monophilic graphs

Abstract

In 1990, Kostochka and Sidorenko proposed studying the smallest number of list-colorings of a graph $G$ among all assignments of lists of a given size $n$ to its vertices. We say a graph $G$ is $n$-monophilic if this number is minimized when identical $n$-color lists are assigned to all vertices of $G$. Kostochka and Sidorenko observed that all chordal graphs are $n$-monophilic for all $n$. Donner (1992) showed that every graph is $n$-monophilic for all sufficiently large $n$. We prove that all cycles are $n$-monophilic for all $n$; we give a complete characterization of 2-monophilic graphs (which turns out to be similar to the characterization of 2-choosable graphs given by Erdos, Rubin, and Taylor in 1980); and for every $n$ we construct a graph that is $n$-choosable but not $n$-monophilic.