Dense Matchings of Linear Size in Graphs with Independence Number 2

For a real number $c > 4$, we prove that every graph $G$ with $α(G) \leq 2$ 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 $α(G) \leq 2$ has $\text{had}(G) \geq (\frac{1}{3} + \varepsilon) |V(G)|$.

💡 Research Summary

The paper addresses a long‑standing question related to Hadwiger’s conjecture in the special case where the independence number α(G) of a graph G is at most two. Hadwiger’s conjecture predicts that the Hadwiger number had(G) (the largest n such that K_n is a minor of G) is at least the chromatic number χ(G). When α(G)≤2, χ(G)≥|V(G)|/2, so the conjecture would imply had(G)≥|V(G)|/2. The best known general bound, due to Duchet and Meyniel, only guarantees had(G)≥|V(G)|/3 in this setting. Seymour asked whether the factor 1/3 can be improved to 1/3+ε for some absolute ε>0. This question is equivalent to the so‑called Linear‑CM conjecture, which asserts that every graph with α(G)≤2 contains a connected matching of linear size.

Instead of proving the existence of a fully connected matching, the author proves a “dense” version: for any real constant c>4 and any integer t≥1, every graph G with α(G)≤2 and at least ct vertices contains a matching M of size t such that the proportion of non‑adjacent pairs of edges inside M is at most

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