Dense Matchings of Linear Size in Graphs with Independence Number 2

TL;DR

This paper proves that graphs with independence number at most 2 and sufficiently many vertices contain large matchings with few non-adjacent pairs, advancing understanding related to Hadwiger's Conjecture.

Contribution

It establishes a new bound on matchings in graphs with independence number 2, connecting to open problems in graph theory and Hadwiger's Conjecture.

Findings

Existence of large matchings with few non-adjacent pairs in graphs with independence number 2

Quantitative bounds related to Seymour's open problem on Hadwiger's Conjecture

Progress towards understanding the structure of dense graphs with restricted independence number

Abstract

For a real number $c > 4$, we prove that every graph $G$ with $\alpha(G) \leq 2$ and $|V(G)| \geq ct$ has a matching $M$ with $|M| = t$ such that the number of non-adjacent pairs of edges in $M$ is at most: \begin{equation*} \left( \frac{1}{c\left(c-1\right)^2} + O_c\left(t^{-1/3} \right) \right) \binom{t}{2}. \end{equation*} This is related to an open problem of Seymour (2016) about Hadwiger's Conjecture, who asked if there is a constant $\varepsilon > 0$ such that every graph $G$ with $\alpha(G) \leq 2$ has $\text{had}(G) \geq (\frac{1}{3} + \varepsilon) |V(G)|$.