Total Equitable List Coloring

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 $k \geq \max\{\chi(T(G)), \Delta(G)+2\}$ 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 $k \geq \max\{\chi_l(T(G)), \Delta(G)+2\}$ where $\chi_l(T(G))$ is the list chromatic number of $T(G)$. We prove this conjecture for all graphs satisfying $\Delta(G) \leq 2$ while also studying the related question of the equitable choosability of powers of paths and cycles.