On the anti-Ramsey threshold

TL;DR

The paper addresses the anti-Ramsey threshold for random graphs with respect to a fixed dense graph $H$. It introduces a novel degenerate decomposition, inspired by the Nine Dragon Tree theorem, that splits any host graph $G$ into a bounded-degree subgraph $B$ and a degenerate remainder, enabling a colouring that precludes rainbow copies of $H$ when $m_2(H)$ is large. The authors prove a deterministic version: if $m(G)\ge 18$, there is a proper edge-colouring in which every rainbow subgraph has $2$-density $d_2$ strictly below $m(G)$, leading to a probabilistic threshold result for anti-Ramsey properties. Consequently, for every strictly $2$-balanced $H$ with $m_2(H)\ge 19$, there exist constants $c_0<c_1$ such that $G_{n,p}$ is anti-Ramsey for $H$ with high probability when $p\ge c_1 n^{-1/m_2(H)}$ and not when $p\le c_0 n^{-1/m_2(H)}$. The decomposition technique may be of independent interest due to its structural control over degeneracy and edge-colouring implications.

Abstract

We say that a graph $G$ is anti-Ramsey for a graph $H$ if any proper edge-colouring of $G$ yields a rainbow copy of $H$, i.e. a copy of $H$ whose edges all receive different colours. In this work we determine the threshold at which the binomial random graph becomes anti-Ramsey for any fixed graph $H$, given that $H$ is sufficiently dense. Our proof employs a graph decomposition lemma in the style of the Nine Dragon Tree theorem that may be of independent interest.

Theorems & Definitions (16)

• Theorem 1: KohKonMot14
• Theorem 2: Nenadov--Person--Škorić--Steger NenPerSkoSte17, Behague--Hancock--Hyde--Letzter--Morrison BehHanHydLetMor24
• Theorem 3: anti-Ramsey threshold
• Theorem 4
• Remark
• Proposition 5: Degenerate decomposition
• Remark
• Lemma 6
• Theorem 7: Fan--Li--Song--Yang FanLiSonYan15
• proof : Proof of Proposition \ref{['prop:degen-decomp']}