Bounds Related to The Edge-List Chromatic and Total Chromatic Numbers of a Simple Graph

Abstract

We show that for a simple graph $G$, $c^{\prime }(G)\le \Delta (G)+2$ where $c^{\prime }(G)$ is the choice index (or edge-list chromatic number) of $G$, and $\Delta (G)$ is the maximum degree of $G$. As a simple corollary of this result, we show that the total chromatic number ${\chi }_T(G)$ of a simple graph satisfies the inequality ${\chi }_T(G)\le \Delta (G)+4$ and the total choice number $c_T(G)$ also satisfies this inequality. We also relate these bounds to the Hall index and the Hall condition index of a simple graph, and to the total Hall number and the total Hall condition number of a simple graph.