On the three graph invariants related to matching of finite simple graphs

Abstract

Let $G$ be a finite simple graph on the vertex set $V(G)$ and let $\text{ind-match}(G)$, $\text{min-match}(G)$ and $\text{match}(G)$ denote the induced matching number, the minimum matching number and the matching number of $G$, respectively. It is known that the inequalities $\text{ind-match}(G) \leq \text{min-match}(G) \leq \text{match}(G) \leq 2\text{min-match}(G)$ and $\text{match}(G) \leq \left\lfloor |V(G)|/2 \right\rfloor$ hold in general. In the present paper, we determine the possible tuples $(p, q, r, n)$ with $\text{ind-match}(G) = p$, $\text{min-match}(G) = q$, $\text{match}(G) = r$ and $|V(G)| = n$ arising from connected simple graphs. As an application of this result, we also determine the possible tuples $(p', q, r, n)$ with ${\rm{reg}}(G) = p'$, $\text{min-match}(G) = q$, $\text{match}(G) = r$ and $|V(G)| = n$ arising from connected simple graphs, where $I(G)$ is the edge ideal of $G$ and ${\rm{reg}}(G) = {\rm{reg}}(K[V(G)]/I(G))$ is the Castelnuovo--Mumford regularity of the quotient ring $K[V(G)]/I(G)$.