Edge-colouring graphs with bounded local degree sums

Abstract

Deciding if a general graph G with maximum degree Δ is Class 1 (Δ-edge-colourable) is an NP-complete problem. Therefore, for over 30 years there has been much work aimed at identifying graph classes wherein Δ-edge-colourability is equivalent to non-subgraph-overfullness, a polynomial-time decidable property. A graph G is said to be subgraph-overfull (SO) if it has a subgraph H with the same maximum degree as G and more than Δ⌊|V(H)|∕2⌋ edges, which is a sufficient condition for G to be Class 2 (non-Δ-edge-colourable). Let X be the class of the graphs whose majors (vertices of degree Δ) have local degree sum at most Δ2−Δ (by ‘local degree sum’ of a vertex u we mean the sum of the degrees of the neighbours of u). The class X is a large graph class, in the sense that almost every graph G is in X, even given that the majors of G induce a graph with cycles. It follows from the definition that all graphs in X are non-SO and we show further that all graphs in X are in fact Class 1. This implies that every vertex of a critical graph (a connected Class 2 graph G such that χ′(G−e)<χ′(G) for every e∈E(G)) is adjacent to at least two majors with local degree sum at least Δ2−Δ+1, extending a well-known result by Vizing of 1965. The proof presented is constructive and yields a polynomial-time edge-colouring algorithm for all graphs in X.