On critical graphs with chromatic index 4

Abstract

It is shown that the number of vertices of valency 2 in a critical graph with chromatic index 4 is at most 1/3 of the total number of vertices, and that there exist critical graphs with just one vertex of valency 2, but none with exactly two vertices of valency 2. From this bounds for the number of edges are deduced. The paper ends with a presentation of a catalogue of all critical graphs with chromatic index 4 and at most 8 vertices, and all simple critical graphs with chromatic index 4 and at most 10 vertices.