On a maximal anti-Ramsey conjecture of Burr, Erdős, Graham, and Sós

Abstract

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 $\left\lfloor n^2/4 \right\rfloor+1\le e\le \binom{n}{2}$.

Figures (4)

• Figure 1: Illustration of how a cycle of length $2k+1$ containing both good edges $xy$ and $zw$ is constructed. Straight lines depict edges, curvy ones denote paths. We have $0\le \ell(P_1) \le 3, \:\ell(P_2)=2k-5-\ell(P_1),$ and $\ell(P_3)=4$.
• Figure 2: Illustration of how a cycle of length $2k+1$ containing both edges $pq$ and $zw$ is constructed. Straight lines depict edges, curvy one denotes a path.
• Figure 3: Illustration of how a cycle of length $2k+1$ containing both edges $pq$ and $pz$ is constructed. Straight lines depict edges, curvy one denotes a path.
• Figure 4: Illustration of how a cycle of length $2k+1$ containing both good edges is constructed. Straight lines depict edges, curvy ones denote paths. We use either the blue edge or the two green edges to adjust the length to precisely $2k+1$.

Theorems & Definitions (15)

• Definition
• Conjecture 1.1: BEGSE
• Theorem 1.2
• Lemma 3.1
• proof
• Claim 1
• proof
• Lemma 3.2
• proof
• Lemma 3.3