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.