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arXiv · 2109.07610

A note on Goldberg's conjecture on total chromatic numbers

Yan Cao·Guantao Chen·Guangming Jing
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Abstract

Let $G=(V(G), E(G))$ be a multigraph with maximum degree $Δ(G)$, chromatic index $χ'(G)$ and total chromatic number $χ''(G)$. The Total Coloring conjecture proposed by Behzad and Vizing, independently, states that $χ''(G)\leq Δ(G)+μ(G) +1$ for a multigraph $G$, where $μ(G)$ is the multiplicity of $G$. Moreover, Goldberg conjectured that $χ''(G)=χ'(G)$ if $χ'(G)\geq Δ(G)+3$ and noticed the conjecture holds when $G$ is an edge-chromatic critical graph. By assuming the Goldberg-Seymour conjecture, we show that $χ''(G)=χ'(G)$ if $χ'(G)\geq \max\{ Δ(G)+2, |V(G)|+1\}$ in this note. Consequently, $χ''(G) = χ'(G)$ if $χ'(G) \ge Δ(G) +2$ and $G$ has a spanning edge-chromatic critical subgraph.

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Yan Cao, Guantao Chen, Guangming Jing. 2021-09-15. A note on Goldberg's conjecture on total chromatic numbers. https://arxiv.org/abs/2109.07610

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