Maximal Induced Matchings in Triangle-Free Graphs

Abstract

An induced matching in a graph is a set of edges whose endpoints induce a 1-regular subgraph. It is known that every n-vertex graph has at most 10n/5≈1.5849n maximal induced matchings, and this bound is the best possible. We prove that every n-vertex triangle-free graph has at most 3n/3≈1.4423n maximal induced matchings; this bound is attained by every disjoint union of copies of the complete bipartite graph K3, 3. Our result implies that all maximal induced matchings in an n-vertex triangle-free graph can be listed in time O(1.4423n), yielding the fastest known algorithm for finding a maximum induced matching in a triangle-free graph.