On graphs decomposable into induced matchings of linear sizes

TL;DR

This paper investigates the maximum number of induced matchings of linear size into graphs, establishing bounds for different ratios and providing tightness results, advancing understanding of graph decompositions.

Contribution

It determines the maximum number of induced matchings in graphs when each has size proportional to the number of vertices, especially for ratios above and at 1/4, with tight bounds and improved upper limits.

Findings

For c > 1/4, maximum t is a constant depending on c.

At c=1/4, t can be as large as Omega(log n).

For fixed c between 1/5 and 1/4, t = O(n/log n), with improvements to o(n/log n).

Abstract

We call a graph $G$ an $(r,t)$-Ruzsa-Szemer\'edi graph if its edge set can be partitioned into $t$ edge-disjoint induced matchings, each of size $r$. These graphs were introduced in 1978 and has been extensively studied since then. In this paper, we consider the case when $r=cn$. For $c>1/4$, we determine the maximum possible $t$ which is a constant depending only on $c$. On the other hand, when $c=1/4$, there could be as many as $\Omega(\log n)$ induced matchings. We prove that this bound is tight up to a constant factor. Finally, when $c$ is fixed strictly between $1/5$ and $1/4$, we give a short proof that the number $t$ of induced matchings is $O(n/\log n)$. We are also able to further improve the upper bound to $o(n/\log n)$ for fixed $c> 1/4-b$ for some positive constant $b$.