Some bound of the edge chromatic surplus of certain cubic graphs

Abstract

V.G. Vizing showed that any graph belongs to one of two classes: Class 1 if χʹ(G) = Δ(G) or in class 2 if χʹ(G) = Δ(G) + 1, where χʹ(G) and Δ(G) denote the edge chromatic index of G and the maximum degree of G, respectively. This paper addresses the problem of finding the edge chromatic surplus of a cubic graph G in Class 2, namely the minimum cardinality of colour classes over all 4-edge chromatic colourings of E(G). An approach to face this problem - using a new parameter q - is given in [1]. Computing q is difficult for the general case of graph G, so there is the need to find restricted classes of G, where q is easy to compute. Working in the same sense as in this paper we give an upper bound of the edge chromatic surplus for a special type of cubic graphs, that is the class of bridgeless non-planar cubic graphs in which in each pair of crossing edges, the crossing edges are adjacent to a third edge.