Partition the vertices of a graph into induced matchings

Abstract

The induced matching partition number of a graph G, denoted by imp( G), is the minimum integer k such that V( G) has a k-partition ( V 1, V 2,…, V k ) such that, for each i, 1⩽ i⩽ k, G[ V i ], the subgraph of G induced by V i , is a 1-regular graph. This is different from the strong chromatic index—the minimum size of a partition of the edges of graph into induced matchings. It is easy to show, as we do in this paper, that, if G is a graph which has a perfect matching, then imp( G)⩽2 Δ( G)−1, where Δ( G) is the maximum degree of a vertex of G. We further show in this paper that, when G is connected, imp( G)=2 Δ( G)−1 if and only if G is isomorphic to either K 2 or C 4 k+2 or the Petersen graph, where C n is the cycle of length n.