The minimum number of vertices and edges of connected graphs with ind-match$(G) = p$, min-match$(G) = q$ and match$(G) = r$

TL;DR

This work investigates extremal vertex and edge counts for connected graphs with fixed ind-match$(G)=p$, min-match$(G)=q$, and match$(G)=r$ within $1\le p\le q\le r\le 2q$. It develops sharp vertex-minima in Theorem 2.1 and a suite of edge-minima results (Theorems 3.1, 3.2, 3.4) supported by explicit constructions (notably the graphs $G_r$ and several $G^{(i)}$ families) to realize the bounds. The paper also leverages two structural conditions (*$_1$) and (*$_2$) to build graphs with prescribed invariant sums, and discusses whiskered and split graphs to connect extremal properties with algebraic combinatorics. An open question about characterizing trees with ind-match$=\,$min-match is posed, highlighting a potential special-case pathway for better understanding the equality $\text{ind-match}(G)=\text{min-match}(G)$ in restricted graph classes. Overall, the results provide explicit, often tight, extremal templates for graphs under coupled matching constraints, with implications for graph theory and related algebraic contexts.

Abstract

Let ind-match$(G)$, min-match$(G)$ and match$(G)$ denote the induced matching number, minimum matching number and matching number of a graph $G$, respectively. It is known that ind-match$(G) \leq $ min-match$(G) \leq {\rm match}(G) \leq$ 2min-match$(G)$ holds. In the present paper, we investigate the minimum number of vertices and edges of connencted simple graphs $G$ with ind-match$(G) = p$, min-match$(G) = q$ and ${\rm match}(G) = r$ for pair of integers $p, q, r$ such that $1 \leq p \leq q \leq r \leq 2q$.

Figures (5)

• Figure 1: The graph $G_{q}^{(1)}$
• Figure 2: The graph $G_{q, r}^{(2)}$
• Figure 3: The graph $G_{r}^{(3)}$
• Figure 4: The graph $G_{q}^{(4)}$
• Figure 5: The graph $G_{q,r}^{(5)}$

Theorems & Definitions (38)

• Definition 1
• Lemma 1.1
• Lemma 1.2: cf. MY
• Lemma 1.3
• proof
• Lemma 1.4
• proof
• Lemma 1.5
• proof
• Definition 1.6