Total Equitable List Coloring

Abstract

An equitable coloring is a proper coloring of a graph such that the sizes of the color classes differ by at most one. A graph G is equitably k-colorable if there exists an equitable coloring of G which uses k colors, each one appearing on either $\lfloor |V(G)|/k \rfloor$ or $\lceil |V(G)|/k \rceil$ vertices of G. In 1994, Fu conjectured that for any simple graph G, the total graph of G, T(G), is equitably k-colorable whenever $k \ge \max \{\chi (T(G)), \Delta (G)+2\}$ where $\chi (T(G))$ is the chromatic number of the total graph of G and $\Delta (G)$ is the maximum degree of G. We investigate the list coloring analogue. List coloring requires each vertex v to be colored from a specified list L(v) of colors. A graph is k-choosable if it has a proper list coloring whenever vertices have lists of size k. A graph is equitably k-choosable if it has a proper list coloring whenever vertices have lists of size k, where each color is used on at most $\lceil |V(G)|/k \rceil$ vertices. In the spirit of Fu’s conjecture, we conjecture that for any simple graph G, T(G) is equitably k-choosable whenever $k \ge \max \{\chi _l(T(G)), \Delta (G)+2\}$ where $\chi _l(T(G))$ is the list chromatic number of T(G). We prove this conjecture for all graphs satisfying $\Delta (G) \le 2$ while also studying the related question of the equitable choosability of powers of paths and cycles.