Dense Matchings of Linear Size in Graphs with Independence Number 2
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866912756004814848 |
|---|---|
| author | Yip, Jung Hon |
| author_facet | Yip, Jung Hon |
| contents | 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 $\text{had}(G) \geq (\frac{1}{3} + \varepsilon) |V(G)|$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_01401 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dense Matchings of Linear Size in Graphs with Independence Number 2 Yip, Jung Hon Combinatorics 05C83 (Primary), 05C15, 05C35 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 $\text{had}(G) \geq (\frac{1}{3} + \varepsilon) |V(G)|$. |
| title | Dense Matchings of Linear Size in Graphs with Independence Number 2 |
| topic | Combinatorics 05C83 (Primary), 05C15, 05C35 |
| url | https://arxiv.org/abs/2512.01401 |