On a maximal anti-Ramsey conjecture of Burr, Erdős, Graham, and Sós
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917353299640320 |
|---|---|
| author | Bucic, Matija Chen, Kaizhe Ma, Jie |
| author_facet | Bucic, Matija Chen, Kaizhe Ma, Jie |
| contents | Given a graph $H$, the maximal anti-Ramsey function $f(n,e,H)$ denotes the minimum integer $f$ for which there exists an $n$-vertex graph $G$ with at least $e$ edges admitting an edge-coloring with $f$ colors in which each copy of $H$ in $G$ is rainbow. In the late 1980s, Burr, Erdős, Graham, and Sós conjectured that for every odd cycle $C_{2k+1}$ with $k \ge 3$, $f(n, \lfloor n^2/4 \rfloor + 1, C_{2k+1}) = n^2/8 + o(n^2)$. In this note, we confirm this conjecture for all $k \ge 4$. More generally, we establish the asymptotic formula $$f\left(n,e,C_{2k+1}\right)=\frac{e}{2}+\frac{n}{2}\sqrt{e-\frac{n^2}{4}}+o(n^2),$$ for the entire non-trivial range of $\left\lfloor n^2/4 \right\rfloor+1\le e\le \binom{n}{2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_18952 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On a maximal anti-Ramsey conjecture of Burr, Erdős, Graham, and Sós Bucic, Matija Chen, Kaizhe Ma, Jie Combinatorics Given a graph $H$, the maximal anti-Ramsey function $f(n,e,H)$ denotes the minimum integer $f$ for which there exists an $n$-vertex graph $G$ with at least $e$ edges admitting an edge-coloring with $f$ colors in which each copy of $H$ in $G$ is rainbow. In the late 1980s, Burr, Erdős, Graham, and Sós conjectured that for every odd cycle $C_{2k+1}$ with $k \ge 3$, $f(n, \lfloor n^2/4 \rfloor + 1, C_{2k+1}) = n^2/8 + o(n^2)$. In this note, we confirm this conjecture for all $k \ge 4$. More generally, we establish the asymptotic formula $$f\left(n,e,C_{2k+1}\right)=\frac{e}{2}+\frac{n}{2}\sqrt{e-\frac{n^2}{4}}+o(n^2),$$ for the entire non-trivial range of $\left\lfloor n^2/4 \right\rfloor+1\le e\le \binom{n}{2}$. |
| title | On a maximal anti-Ramsey conjecture of Burr, Erdős, Graham, and Sós |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.18952 |