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Maximal and maximum induced matchings in connected graphs

Bo-Jun Yuan, Zhao-Yu Yang, Lu Zheng, Shi-Cai Gong

Abstract

An induced matching in a graph is a set of edges whose endpoints induce a $1$-regular subgraph. Gupta et al. (2012,\cite{Gupta}) showed that every $n$-vertex graph has at most $10^{\frac{n}{5}}\approx 1.5849^n$ maximal induced matchings, which is attained by the disjoint union of copies of the complete graph $K_5$. In this paper, we show that the maximum number of maximal and maximum induced matchings in a connected graph of order $n$ is \begin{align*} \begin{cases} {n\choose 2} &~ {\rm if}~ 1\leq n\le 8; \\ {{\lfloor \frac{n}{2} \rfloor}\choose 2}\cdot {{\lceil \frac{n}{2} \rceil}\choose 2} -(\lfloor \frac{n}{2} \rfloor-1)\cdot (\lceil \frac{n}{2} \rceil-1)+1 &~ {\rm if}~ 9\leq n\le 13; \\ 10^{\frac{n-1}{5}}+\frac{n+144}{30}\cdot 6^{\frac{n-6}{5}} &~ {\rm if}~ 14\leq n\le 30;\\ 10^{\frac{n-1}{5}}+\frac{n-1}{5}\cdot 6^{\frac{n-6}{5}} & ~ {\rm if}~ n\geq 31, \\ \end{cases} \end{align*} and also show that this bound is tight. This result implies that we can enumerate all maximal induced matchings of an $n$-vertex connected graph in time $O(1.5849^n)$. Moreover, our result provides an estimate on the number of maximal dissociation sets of an $n$-vertex connected graph.

Maximal and maximum induced matchings in connected graphs

Abstract

An induced matching in a graph is a set of edges whose endpoints induce a -regular subgraph. Gupta et al. (2012,\cite{Gupta}) showed that every -vertex graph has at most maximal induced matchings, which is attained by the disjoint union of copies of the complete graph . In this paper, we show that the maximum number of maximal and maximum induced matchings in a connected graph of order is \begin{align*} \begin{cases} {n\choose 2} &~ {\rm if}~ 1\leq n\le 8; \\ {{\lfloor \frac{n}{2} \rfloor}\choose 2}\cdot {{\lceil \frac{n}{2} \rceil}\choose 2} -(\lfloor \frac{n}{2} \rfloor-1)\cdot (\lceil \frac{n}{2} \rceil-1)+1 &~ {\rm if}~ 9\leq n\le 13; \\ 10^{\frac{n-1}{5}}+\frac{n+144}{30}\cdot 6^{\frac{n-6}{5}} &~ {\rm if}~ 14\leq n\le 30;\\ 10^{\frac{n-1}{5}}+\frac{n-1}{5}\cdot 6^{\frac{n-6}{5}} & ~ {\rm if}~ n\geq 31, \\ \end{cases} \end{align*} and also show that this bound is tight. This result implies that we can enumerate all maximal induced matchings of an -vertex connected graph in time . Moreover, our result provides an estimate on the number of maximal dissociation sets of an -vertex connected graph.

Paper Structure

This paper contains 8 sections, 39 theorems, 83 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['main1']}
  4. Property A: $1<d(u)<n-1$ holds for any vertex $u$ of a minimal counterexample
  5. Property B: $u$ is a cutpoint if $d(u)\ge 6$
  6. Property C: each minimal counterexample has no pendant blocks $K_3, K_4, K_5, K_6$
  7. Property D: each minimal counterexample has no non-cutpoints
  8. Concluding remarks

Key Result

Proposition 1.1

Gupta Let $G$ be a (not necessarily connected) graph of order $n$. Then $|M_G|\leq 10^{\frac{n}{5}}$ with equality iff $n\equiv 0~ (\bmod~5)$ and $G= \frac{n}{5}K_5$.

Figures (2)

  • Figure 1: $K_r\ast (K_{s_1}\cup K_{s_2}\cup \cdots \cup K_{s_t})$.
  • Figure 2: Graphs $G$ (left) and $G_{v\rightarrow u}$ (right).

Theorems & Definitions (72)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Remark 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 2.1
  • proof
  • ...and 62 more