Ring graphs and Goldberg's bound on chromatic index

Let G be a multigraph with maximum degree Δ and chromatic index χ′. If G is bipartite then χ′ = Δ. Otherwise, by a theorem of Goldberg, χ′ ≤ Δ + 1 + ⌊(Δ − 2)/(g_o − 1)⌋, where g_o denotes the odd girth of G. Stiebitz, Scheide, Toft, and Favrholdt in their book conjectured that if χ′ = Δ + 1 + ⌊(Δ − 2)/(go − 1)⌋ then G contains as a subgraph a ring graph R with the same chromatic index. Vizing's characterization of graphs with chromatic index attaining the Shannon's bound showed the above conjecture holds for g_o = 3. Stiebitz et al verified the conjecture for graphs with g_o = 5 and Δ ≥ 10. McDonald proved the conjecture when Δ − 2 is divisible by g_o − 1. In this paper, we show that the chromatic index condition alone is not sufficient to give the conclusion in the conjecture. On the positive side, we show that the conjecture holds for every go ≥ 3 with Δ ≥ (go − 2go + 5)/2, and the maximum degree condition is best possible. This positive result leans on the positive resolution of the Goldberg-Seymour conjecture.