A sufficient condition for a regular graph to be class 1

The core GΔ of a simple graph G is the subgraph induced by the vertices of maximum degree. It is well known that the Petersen graph is not 1-factorizable and has property that the core of the graph obtained from it by removing one vertex has maximum degree 2. In this paper, we prove the following result. Let G be a regular graph of even order with d(G) ≥ 3. Suppose that G contains a vertex ν such that the core of G\ν has maximum degree 2. If G is not the Petersen graph, then G is 1-factorizable. © 1993 John Wiley & Sons, Inc.