Reducible 3-critical graphs

In this paper, we provide further insights into the reducibility of 3-critical graphs. A graph is 3-critical if it is class two, that is, has chromatic index 4, and the removal of any one edge renders a graph with chromatic index 3. We consider two types of reductions: the suppression of a 2-connected subgraph into a single edge; and the suppression of a 3-connected subgraph into a single vertex. That is, in cases where the 2- or 3-connected subgraph is cubic or strictly subcubic. We show that every cyclically 2-connected 3-critical graph can be reduced to a smaller 3-critical graph using this method. We also prove that every 3-critical graph which is cyclically k -connected with k = 2 or k = 3, contains a 3-critical minor which is cyclically k -connected with k ≥ 4.