Dominating induced matchings of finite graphs and regularity of edge ideals

TL;DR

This paper explores the relationship between dominating induced matchings in finite graphs and the regularity of their edge ideals, linking combinatorial properties with algebraic invariants.

Contribution

It characterizes graphs with dominating induced matchings and establishes conditions under which the induced matching number equals the minimum matching number.

Findings

Graphs with dominating induced matchings have equal induced and minimum matching numbers.

The regularity of edge ideals is bounded by matching numbers, with equality in special cases.

The study bridges combinatorial graph properties and algebraic invariants of edge ideals.

Abstract

The regularity of an edge ideal of a finite simple graph $G$ is at least the induced matching number of $G$ and is at most the minimum matching number of $G$. If $G$ possesses a dominating inuduced matching, i.e., an induced matching which forms a maximal matching, then the induced matching number of $G$ is equal to the minimum matching number of $G$. In the present paper, from viewpoints of both combinatorics and commutative algebra, finite simple graphs with dominating induced matchings will be mainly studied.