Achieving maximum chromatic index in multigraphs

Let G be a multigraph with maximum degree Δ and maximum edge multiplicity μ. Vizing’s Theorem says that the chromatic index of G is at most Δ+μ. If G is bipartite its chromatic index is well known to be exactly Δ. Otherwise G contains an odd cycle and, by a theorem of Goldberg, its chromatic index is at most , where g_o denotes odd-girth. Here we prove that a connected G achieves Goldberg’s upper bound if and only if G=μC_go and (g_o−1)∣2(μ−1). The question of whether or not G achieves Vizing’s upper bound is NP-hard for μ=1, but for μ≥2 we have reason to believe that this may be answerable in polynomial time. We prove that, with the exception of μK₃, every connected G with μ≥2 which achieves Vizing’s upper bound must contain a specific dense subgraph on five vertices. Additionally, if Δ≤μ², we prove that G must contain K₅, so G must be nonplanar. These results regarding Vizing’s upper bound extend work by Kierstead, whose proof technique influences us greatly here.