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

TL;DR

This paper characterizes all possible combinations of three graph invariants related to matchings in connected simple graphs and explores their relation to algebraic properties like Castelnuovo–Mumford regularity.

Contribution

It determines the feasible tuples of induced matching, minimum matching, and matching numbers, and links these to algebraic invariants for connected simple graphs.

Findings

Characterizes all possible invariant tuples for connected simple graphs.

Establishes relations between matching invariants and algebraic regularity.

Provides a comprehensive classification of these invariants in graph theory and algebra.

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 $K[V(G)]/I(G)$.