Some Results on Dominating Induced Matchings

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 χ ( G ) ≤ 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 2 r - 1 r - 1 and n = 2 r - 1 r - 1 if and only if G is the Kneser graph with parameters r - 1 , 2 r - 1 .