The average size of maximal matchings in graphs

Abstract

We investigate the ratio $\avM(G)$ of the average size of a maximal matching to the size of a maximum matching in a graph $G$. If many maximal matchings have a size close to $\maxM(G)$, this graph invariant has a value close to 1. Conversely, if many maximal matchings have a small size, $\avM(G)$ approaches $\frac{1}{2}$. We propose a general technique to determine the asymptotic behavior of $\avM(G)$ for various classes of graphs. To illustrate the use of this technique, we first show how it makes it possible to find known asymptotic values of $\avM(G)$ which were typically obtained using generating functions, and we then determine the asymptotic value of $\avM(G)$ for other families of graphs, highlighting the spectrum of possible values of this graph invariant between $\frac{1}{2}$ and $1$.

Figures (13)

• Figure 1: $\widetilde{{\sf P}}_{4}$ and $\widetilde{{\sf K}}_{3}$.
• Figure 2: A comparison of $\mathcal{I}(G)$, $\mathcal{I}^{o}(G)$ and $\mathcal{I}^{DF}(G)$ on all graphs of order 10.
• Figure 3: Two graphs $G_1$ and $G_2$ with $\mathcal{I}(G_1)>\mathcal{I}^{DF}(G_1)$ and $\mathcal{I}^{o}(G_2)<\mathcal{I}^{DF}(G_2)$.
• Figure 4: $G_{3}^{1,1}$, $G_{3}^{1,2}$ and $G_{3}^{1,3}$.
• Figure 5: $G_{n}^{1,2,1}$, $G_{n}^{1,2,2}$ and $G_{n}^{1,2,3}$.

Theorems & Definitions (32)

• Theorem 1
• proof
• Corollary 2
• proof
• Theorem 3
• proof
• Corollary 4
• proof
• Corollary 5
• proof