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

TL;DR

This paper establishes upper bounds for the edge-list chromatic number and total chromatic number of simple graphs, linking these bounds to classical graph invariants and providing new insights into graph coloring constraints.

Contribution

It proves that the choice index is at most two above the maximum degree and that the total chromatic number is at most four above the maximum degree, connecting these bounds to Hall-type parameters.

Findings

c'(G)  \u2264 (G)+2

_T(G)  (G)+4

Bounds relate to Hall index and Hall condition parameters

Abstract

We show that for a simple graph $G$, $c'(G)\leq\Delta(G)+2$ where $c'(G)$ 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 $\chi_T(G)\leq\ \Delta(G)+4$ 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.