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### Title: 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 $\chi(G) \leqslant 3$. 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 $\binom{2r - 1}{r - 1}$ and $n = \binom{2r - 1}{r - 1}$ if and only if $G$ is the Kneser graph with parameters $r-1$, $2r-1$.

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