Rainbow subgraphs of uniformly coloured randomly perturbed graphs

TL;DR

This work studies rainbow copies of bounded-degree trees in uniformly coloured randomly perturbed graphs $G_0\cup \mathbf{G}(n,p)$ with $G_0\in\mathcal{G}_{\delta,n}$. It develops a McDiarmid-type coupling to convert uncoloured embedding results into rainbow ones, yielding a general mechanism for finding rainbow subgraphs in perturbed graphs when colours are up to $(1+o(1))e(H)$; this recovers and extends rainbow Hamilton cycles and bounded-degree rainbow trees in the perturbed setting. The authors prove that for any $\delta\in(0,1)$ and integer $d\ge 2$, there exists $C=C(\delta,d)$ such that a uniformly coloured $G_0\cup \mathbf{G}(n,C/n)$ with colours in $[n-1]$ contains a rainbow copy of any $n$-vertex tree $T$ with $\Delta(T)\le d$ with high probability, achieving optimal colour and edge-probability up to constants. The approach combines almost-spanning rainbow embeddings in sparse random subgraphs of pseudorandom hosts with absorbers and template methods to handle trees with many leaves or many bare paths, and it yields sharp thresholds in the coloured perturbed model. These results sharpen our understanding of rainbow structures in perturbed graphs and open avenues toward rainbow universality questions.

Abstract

For a given $δ\in (0,1)$, the randomly perturbed graph model is defined as the union of any $n$-vertex graph $G_0$ with minimum degree $δn$ and the binomial random graph $\mathbf{G}(n,p)$ on the same vertex set. Moreover, we say that a graph is uniformly coloured with colours in $\mathcal{C}$ if each edge is coloured independently and uniformly at random with a colour from $\mathcal{C}$. Based on a coupling idea of McDiarmird, we provide a general tool to tackle problems concerning finding a rainbow copy of a graph $H=H(n)$ in a uniformly coloured perturbed $n$-vertex graph with colours in $[(1+o(1))e(H)]$. For example, our machinery easily allows to recover a result of Aigner-Horev and Hefetz concerning rainbow Hamilton cycles, and to improve a result of Aigner-Horev, Hefetz and Lahiri concerning rainbow bounded-degree spanning trees. Furthermore, using different methods, we prove that for any $δ\in (0,1)$ and integer $d \ge 2$, there exists $C=C(δ,d)>0$ such that the following holds. Let $T$ be a tree on $n$ vertices with maximum degree at most $d$ and $G_0$ be an $n$-vertex graph with $δ(G_0)\ge δn$. Then a uniformly coloured $G_0 \cup \mathbf{G}(n,C/n)$ with colours in $[n-1]$ contains a rainbow copy of $T$ with high probability. This is optimal both in terms of colours and edge probability (up to a constant factor).

Figures (1)

• Figure 1: The top figure shows a $(v,c)$-gadget $A_{v,c}$, where we remark that $\mathcal{C}(w_1w_4)=c$ and all colours are pairwise different, except for $\mathcal{C}(vu_1)=\mathcal{C}(w_1w_2)$, $\mathcal{C}(vu_2)=\mathcal{C}(w_3w_4)$ and $\mathcal{C}(u_1u_2)=\mathcal{C}(w_2w_3)$. The bottom-left figure highlights the absorbing path $P$ and the bottom-right one highlights the avoiding path $P'$.

Theorems & Definitions (40)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• proof : Proof of \ref{['thm:mc_diarmid_argument']}
• Definition 3.1: Pseudorandom graph
• Theorem 3.2
• Proposition 3.3
• Theorem 3.4: Consequence of ferber-kriv
• proof : Proof of \ref{['thm:rainbow-almost-spanning-embedding']}