Induced matching extendable graphs

Abstract

We say that a simple graph G is induced matching extendable, shortly IM-extendable, if every induced matching of G is included in a perfect matching of G. The main results of this paper are as follows: (1) For every connected IM-extendable graph G with |V(G)| ≥ 4, the girth g(G) ≤ 4. (2) If G is a connected IM-extendable graph, then |E(G)| ≥ ${3\over 2}|V(G)| - 2$; the equality holds if and only if G ≅ T × K2, where T is a tree. (3) The only 3-regular connected IM-extendable graphs are Cn × K2, for n ≥ 3, and C2n(1, n), for n ≥ 2, where C2n(1, n) is the graph with 2n vertices x0, x1, …, x2n−1, such that xixj is an edge of C2n(1, n) if either |i − j| ≡ 1 (mod 2n) or |i − j| ≡ n (mod 2n). © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 203–213, 1998