Some Results on Dominating Induced Matchings

TL;DR

This paper investigates the properties of graphs with dominating induced matchings (DIM), establishing bounds on chromatic number, characterizing certain graph classes, and linking edge partitions into DIM with specific regular graphs and Kneser graphs.

Contribution

It provides new characterizations of graphs with DIM, bounds on chromatic number, and connects edge partitions into DIM with regular and Kneser graphs.

Findings

If a graph has a DIM, then its chromatic number is at most 3.

Graphs with all edges partitionable into DIM are either regular or biregular.

For r-regular graphs partitionable into DIM, the number of vertices is divisible by a specific binomial coefficient, with equality characterizing Kneser graphs.

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 $\chi(G) \leqslant 3$. 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 $r-1$, $2r-1$.