The Overfullness of Graphs with Small Minimum Degree and Large Maximum Degree

Given a simple graph G, denote by ∆(G), δ(G), and χ(G) the maximum degree, the minimum degree, and the chromatic index of G, respectively. We say G is ∆-critical if χ(G) = ∆(G) + 1 and χ(H) ≤ ∆(G) for every proper subgraph H of G; and G is overfull if |E(G)| > ∆⌊|V (G)|/2⌋. Since a maximum matching in G can have size This work was supported in part by NSF grant DMS-1855716. This work was supported in part by NSF grant DMS-2001130.