Vertex-splitting and Chromatic Index Critical Graphs

We study graphs which are critical with respect to the chromatic index. We relate these to the Overfull Conjecture and we study in particular their construction from regular graphs by subdividing an edge or by splitting a vertex. In this paper, we consider simple graphs (that is graphs which have no loops or multiple edges). An edge-colouring of a graph G is a map 4 : E(G) -+ cp, where cp is a set of colours and E(G) is the set of edges of G, such that no two incident edges receive the same colour. The chromatic index, x’(G) of G is the least value of 1~1 for which an edge-colouring of G exists. A well-known theorem of Vizing [13] states that, for a simple graph G, d(G) 6 x’(G) < d(G) + 1, where d(G) denotes the maximum degree of G. Graphs for which x’(G) = d(G) are said to be Class 1, and otherwise they are Class 2. A graph G is critical if it is Class 2, connected and for each edge e of G, X’(G\e) < x’(G). A fairly long-standing problem has been the attempt to classify which graphs are Class 1, and which graphs are Class 2. Holyer [lo] showed that the problem of determining whether a graph is Class 1 is NP-hard. Notwithstanding this, the Overfull Conjecture of Chetwynd and Hilton [3], if true, would classify all graphs satisfying d(G) > f 1 V(G)1 into Class 1 and Class 2 graphs. A graph is called overfull if * Corresponding author. E-mail: [email protected]. Also, Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA. 0166-218X/97/$17.00