On the chromatic index of multigraphs without large triangles

Let M be a multigraph. Vizing (Kibernetika (Kiev) 1 (1965), 29-39) proved that χ′(M)≤Δ(M)+μ(M). Here it is proved that if χ′(M)≥Δ(M)+s, where 1 2(μ(M) + 1) < s then M contains a 2s-sided triangle. In particular, (C′) if μ(M)≤2 and M does not contain a 4-sided triangle then χ′(M)≤Δ(M) + 1. Javedekar (J. Graph Theory 4 (1980), 265-268) had conjectured that (C) if G is a simple graph that does not induce K_1,3 or K₅-e then χ(G)≤ω(G) + 1. The author and Schmerl (Discrete Math. 45 (1983), 277-285) proved that (C′) implies (C); thus Javedekar's conjecture is true.