Recent progress on edge-colouring graphs (Q1090683)

Suppose G is a multigraph without loops. Let \(\chi\) '(G) and \(\Delta\) (G) denote respectively the chromatic index and the maximum degree of G. If \(\chi '(G)=\Delta (G)\), then G is said to be Class 1, and otherwise G is Class 2. A graph G is said to be overfull, if \(| E(G)| >\Delta (G)\lfloor | V(G)| \rfloor.\) This paper summarizes some of the progress made recently by the author, A. G. Chetwynd and P. D. Johnson about \(\chi\) '(G) with relatively large \(\Delta\) (G). It also mentions seven conjectures of which the most interesting one is Conjecture 1. If G is a simple graph with \(\Delta (G)>\frac{1}{3}| V(G)|\), then G is Class 2 if and only if G contains an overfull subgraph H with \(\Delta (H)=\Delta (G).\) Conjecture 1 implies Conjectures 2-5 and Conjecture 7.