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 Δ + 1 + Δ − 2 g o − 1 , where g o denotes odd-girth. Here we prove that a connected G achieves Goldberg’s upper bound if and only if G = μ C g o 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 μ K 3 , every connected G with μ ≥ 2 which achieves Vizing’s upper bound must contain a specific dense subgraph on five vertices. Additionally, if Δ ≤ μ 2 , we prove that G must contain K 5 , so G must be nonplanar. These results regarding Vizing’s upper bound extend work by Kierstead, whose proof technique influences us greatly here.