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

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)≤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)≥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)≥max{Δ(G)+2,∣V(G)∣+1} in this note. Consequently, χ″(G)=χ′(G) if χ′(G)≥Δ(G)+2 and G has a spanning edge‐chromatic critical subgraph. [ABSTRACT FROM AUTHOR]