Multigraphs with $\Delta \ge 3$ are Totally-$(2\Delta-1)$-choosable

TL;DR

This paper proves that the total graph of a multigraph with maximum degree Δ is (2Δ-1)-choosable and provides a simple, linear-time coloring algorithm that improves or matches previous bounds for various Δ values.

Contribution

It establishes a new upper bound for total graph choosability and introduces a simpler, efficient linear-time coloring algorithm that outperforms previous methods in certain cases.

Findings

Total graphs with maximum degree Δ are (2Δ-1)-choosable.

The proposed algorithm runs in linear time.

Improves bounds for Δ=4 and matches bounds for Δ=3,5,6.

Abstract

The \emph{total graph} $T(G)$ of a multigraph $G$ has as its vertices the set of edges and vertices of $G$ and has an edge between two vertices if their corresponding elements are either adjacent or incident in $G$. We show that if $G$ has maximum degree $\Delta(G)$, then $T(G)$ is $(2\Delta(G)-1)$-choosable. We give a linear-time algorithm that produces such a coloring. The best previous general upper bound for $\Delta(G) > 3$ was $\floor{\frac32\Delta(G)+2}$, by Borodin et al. When $\Delta(G)=4$, our algorithm gives a better upper bound. When $\Delta(G)\in\{3,5,6\}$, our algorithm matches the best known bound. However, because our algorithm is significantly simpler, it runs in linear time (unlike the algorithm of Borodin et al.).