Minimum maximal matchings in cubic graphs

TL;DR

This paper proves new upper bounds on the size of maximal matchings in cubic and degree-3 graphs, confirming a conjecture and providing efficient algorithms with approximation guarantees.

Contribution

It establishes the cubic case of a conjecture on maximal matchings and offers an efficient method to find such matchings with near-optimal size.

Findings

Maximal matching size in connected cubic graphs is at most (5/12)n + 1/2.

General bound for graphs with max degree 3: at most (4n - m)/6 + 1/2.

Efficient algorithms can find these maximal matchings, leading to approximation guarantees.

Abstract

We prove that every connected cubic graph with $n$ vertices has a maximal matching of size at most $\frac{5}{12} n+ \frac{1}{2}$. This confirms the cubic case of a conjecture of Baste, F\"urst, Henning, Mohr and Rautenbach (2019) on regular graphs. More generally, we prove that every graph with $n$ vertices and $m$ edges and maximum degree at most $3$ has a maximal matching of size at most $\frac{4n-m}{6}+ \frac{1}{2}$. These bounds are attained by the graph $K_{3,3}$, but asymptotically there may still be some room for improvement. Moreover, the claimed maximal matchings can be found efficiently. As a corollary, we have a $\left(\frac{25}{18} + O \left( \frac{1}{n}\right)\right) $-approximation algorithm for minimum maximal matching in connected cubic graphs.