# A note on Goldberg's conjecture on total chromatic numbers

@article{Cao2021ANO, title={A note on Goldberg's conjecture on total chromatic numbers}, author={Yan Cao and Guantao Chen and Guangming Jing}, journal={Journal of Graph Theory}, year={2021} }

Let G = (V (G), E(G)) be a multigraph with maximum degree ∆(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) ≤ ∆(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) ≥ ∆(G) + 3 and noticed the conjecture holds when G is an edge-chromatic critical graph. By assuming the Goldberg-Seymour conjecture, we show that… Expand

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