[Paper Review] Universality of random graphs and rainbow embedding

Asaf Ferber, Rajko Nenadov+1 more|arXiv (Cornell University)|Nov 27, 2013

Limits and Structures in Graph Theory19 references4 citations

TL;DR

This paper establishes improved threshold probabilities for the universality of random graphs $γ(n,p)$ in embedding spanning graphs with bounded maximum degree and density, and introduces a novel random edge-coloring model to guarantee rainbow embeddings. It proves that for $p = \omega(\Delta^{12}n^{-1/2}\log^3 n)$, a typical random graph contains all spanning trees with maximum degree $\Delta$, and for $p = \omega(\Delta^{12}n^{-1/d}\log^3 n)$, it embeds all such graphs with girth at least 7. Furthermore, it shows that with $(1+o(1))|E(H)|$ random colors, a rainbow copy of any constant-degree $H$ is w.h.p. present.

ABSTRACT

In this paper we show how to use simple partitioning lemmas in order to embed spanning graphs in a typical member of $G(n,p)$. Let the \emph{maximum density} of a graph $H$ be the maximum average degree of all the subgraphs of $H$. First, we show that for $p=ω(Δ^{12} n^{-1/2d}\log^3n)$, a graph $G\sim G(n,p)$ w.h.p.\ contains copies of all spanning graphs $H$ with maximum degree at most $Δ$ and maximum density at most $d$. For $d

Motivation & Objective

• To improve the threshold probability $p$ for which $\mathcal{G}(n,p)$ w.h.p. contains all spanning graphs $H$ with maximum degree $\leq \Delta$ and maximum density $\leq d$.
• To strengthen embedding results for graphs with girth at least 7, particularly spanning trees, by reducing the required edge probability $p$.
• To establish conditions under which a randomly edge-colored $\mathcal{G}(n,p)$ contains a rainbow copy of any constant-degree spanning graph $H$.
• To develop a new embedding technique using partitioning lemmas and color-aware matching processes in random graphs with random edge coloring.

Proposed method

• Utilizes simple graph partitioning lemmas to decompose the host graph and manage embedding constraints.
• Applies a matching-based embedding framework inspired by Alon and Füredi and Ruciński, adapted to handle degree and density constraints.
• Introduces a two-phase embedding process: Phase I embeds a core subgraph using a bounded-degree matching construction; Phase II extends it using a color-aware matching process.
• Employs Chernoff bounds to control concentration of neighbor sets and color availability in the random coloring model.
• Uses a random ordering of vertices and dynamic color set reduction to ensure distinct colors in the final embedding.
• Samples from a structured family of bipartite graphs $\mathcal{B}_{\lceil \log^2 n \rceil\text{-out}}^\ell(F)$ uniformly at random during embedding to maintain color diversity.

Experimental results

Research questions

• RQ1 What is the optimal edge probability $p$ such that $\mathcal{G}(n,p)$ w.h.p. contains all spanning graphs with maximum degree $\leq \Delta$ and maximum density $\leq d$?
• RQ2 Can the threshold for universality be improved when restricting to graphs with girth at least 7, especially for spanning trees?
• RQ3 Under what conditions does a randomly edge-colored $\mathcal{G}(n,p)$ w.h.p. contain a rainbow copy of a given constant-degree spanning graph $H$?
• RQ4 Can a systematic embedding method be designed that simultaneously respects degree constraints, density bounds, and color uniqueness in random edge-coloring models?

Key findings

• For $p = \omega(\Delta^{12}n^{-1/2}\log^3 n)$, $\mathcal{G}(n,p)$ is w.h.p. universal for all spanning trees with maximum degree $\leq \Delta$, improving upon Johannsen, Krivelevich, and Samotij's bound.
• For $p = \omega(\Delta^{12}n^{-1/d}\log^3 n)$, $\mathcal{G}(n,p)$ is w.h.p. universal for all spanning graphs with maximum degree $\leq \Delta$, maximum density $\leq d$, and girth at least 7.
• The threshold $p = \omega(\Delta^{12}n^{-1/2}\log^3 n)$ is optimal up to logarithmic factors for spanning tree universality in $\mathcal{G}(n,p)$.
• For random edge-coloring with $c = (1+o(1))|E(H)|$ colors, a rainbow copy of any constant-degree $H$ is w.h.p. embeddable in $\mathcal{G}(n,p)$ when $p = \omega(\Delta^{12}n^{-1/2}\log^3 n)$.
• The embedding process succeeds with high probability due to concentration bounds on neighbor sets and color availability, ensuring $\Omega(\log^3 n)$ valid extensions at each step.
• The process uniformly samples from a structured family of bipartite graphs, ensuring that the final matching is both rainbow and valid under color constraints.