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

We show that for a simple graph $G$, $c'(G)\leqΔ(G)+2$ where $c'(G)$ is the choice index (or edge-list chromatic number) of $G$, and $Δ(G)$ is the maximum degree of $G$. As a simple corollary of this result, we show that the total chromatic number $χ_T(G)$ of a simple graph satisfies the inequality $χ_T(G)\leq\ Δ(G)+4$ 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.