Rainbow copies of spanning subgraphs

TL;DR

This paper investigates the probabilistic thresholds for the existence of rainbow-colored spanning subgraphs in randomly edge-colored graphs, providing bounds on parameters ensuring such structures appear with high probability.

Contribution

It establishes new lower bounds on edge probability and color set size needed for rainbow spanning subgraphs in random graphs.

Findings

Derived bounds for edge probability p ensuring rainbow copies

Established minimum color set size κ for spanning subgraphs

Proved high probability existence of rainbow subgraphs under these bounds

Abstract

Let $G_{n,p}^{[\kappa]}$ denote the space of $n$-vertex edge coloured graphs, where each edge occurs independently with probability $p$. The colour of each existing edge is chosen independently and uniformly at random from the set $[\kappa]$. We consider the threshold for the existence of rainbow colored copies of a spanning subgraph $H$. We provide lower bounds on $p$ and $\kappa$ sufficient to prove the existence of such copies w.h.p.