On critical graphs with chromatic index 4

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 ...

Chromatic-index-critical graphs of orders 13 and 14

A graph is chromatic-index-critical if it cannot be edge-coloured with ∆ colours (with ∆ the maximal degree of the graph), and if the removal of any edge decreases its chromatic index. The Critical Graph Conjecture stated that any such graph has odd order. It has been proved false and the smallest known counterexample has order 18 [18, 31]. In this paper we show that there are no chromatic-inde...

On the Delta-subgraph of graphs which are critical with respect to the chromatic index

A Class of Edge Critical 4-Chromatic Graphs

We consider several constructions of edge critical 4-chromatic graphs which can be written as the union of a bipartite graph and a matching. In particular we construct such a graph G with each of the following properties: G can be contracted to a given critical 4-chromatic graph; for each n;,: 7, G has n vertices and three matching edges (it is also 8n ) shown that such graphs must have at leas...

Game chromatic index of graphs with given restrictions on degrees

Given a graph G and an integer k, two players alternatively color the edges of G using k colors so that adjacent edges get different colors. The game chromatic index χg(G) is the minimum k for which the first player has a strategy that ensures that all edges of G get colored. The trivial bounds are ∆(G) ≤ χg(G) ≤ 2∆(G)−1, where ∆(G) denote the maximal degree of G. Lam, Shiu, and Xu and, indepen...