Graphs which are vertex-critical with respect to the edge-chromatic number

Abstract

In this paper, multigraphs will have no loops. For a multigraph G , the least number of colours needed to colour the edges of G in such a way that no two edges on the same vertex of G have the same colour, is called the edge-chromatic number , or the chromatic index , of G , and denoted χ′( G ). It is clear that if Δ( G ) denotes the maximum degree of G , then Δ( G ) ≤ χ′( G ). If Δ( G ) = χ′( G ), then G is Class 1, and otherwise G is Class 2.