Some Results on Dominating Induced Matchings

Abstract

Let G be a graph, a dominating induced matching (DIM) of G is an induced matching that dominates every edge of G. In this paper we show that if a graph G has a DIM, then \(\chi (G) \le 3\). Also, it is shown that if G is a connected graph whose all edges can be partitioned into DIM, then G is either a regular graph or a biregular graph and indeed we characterize all graphs whose edge set can be partitioned into DIM. Also, we prove that if G is an r-regular graph of order n whose edges can be partitioned into DIM, then n is divisible by \(\left( {\begin{array}{c}2r - 1\\ r - 1\end{array}}\right) \) and \(n = \left( {\begin{array}{c}2r - 1\\ r-1\end{array}}\right) \) if and only if G is the Kneser graph with parameters \(r-1\), \(2r-1\).