Path-monochromatic bounded depth rooted trees in (random) tournaments
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866916194756329472 |
|---|---|
| author | Yuster, Raphael |
| author_facet | Yuster, Raphael |
| contents | An edge-colored rooted directed tree (aka arborescence) is path-monochromatic if every path in it is monochromatic. Let $k,\ell$ be positive integers. For a tournament $T$, let $f_T(k)$ be the largest integer such that every $k$-edge coloring of $T$ has a path-monochromatic subtree with at least $f_T(k)$ vertices and let $f_T(k,\ell)$ be the restriction to subtrees of depth at most $\ell$. It was proved by Landau that $f_T(1,2)=n$ and proved by Sands et al. that $f_T(2)=n$ where $|V(T)|=n$. Here we consider $f_T(k)$ and $f_T(k,\ell)$ in more generality, determine their extremal values in most cases, and in fact in all cases assuming the Caccetta-Häggkvist Conjecture. We also study the typical value of $f_T(k)$ and $f_T(k,\ell)$, i.e., when $T$ is a random tournament. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_03752 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Path-monochromatic bounded depth rooted trees in (random) tournaments Yuster, Raphael Combinatorics 05C35, 05C20 An edge-colored rooted directed tree (aka arborescence) is path-monochromatic if every path in it is monochromatic. Let $k,\ell$ be positive integers. For a tournament $T$, let $f_T(k)$ be the largest integer such that every $k$-edge coloring of $T$ has a path-monochromatic subtree with at least $f_T(k)$ vertices and let $f_T(k,\ell)$ be the restriction to subtrees of depth at most $\ell$. It was proved by Landau that $f_T(1,2)=n$ and proved by Sands et al. that $f_T(2)=n$ where $|V(T)|=n$. Here we consider $f_T(k)$ and $f_T(k,\ell)$ in more generality, determine their extremal values in most cases, and in fact in all cases assuming the Caccetta-Häggkvist Conjecture. We also study the typical value of $f_T(k)$ and $f_T(k,\ell)$, i.e., when $T$ is a random tournament. |
| title | Path-monochromatic bounded depth rooted trees in (random) tournaments |
| topic | Combinatorics 05C35, 05C20 |
| url | https://arxiv.org/abs/2404.03752 |