A note on Goldberg's conjecture on total chromatic numbers. Issue 1 (10th November 2021) - Cao, Yan; Chen, Guantao; Jing, Guangming
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 ) ≤ 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.