Rainbow copies of spanning subgraphs

2025-05-27

AI Summary

This study determines the threshold for the existence of rainbow spanning subgraphs—subgraphs isomorphic to a given $n$-vertex spanning graph $H$ and having all edges assigned distinct colors—in the randomly edge-colored Erdős–Rényi graph $G_{n,p}^{[kappa]}$. The central problem is to identify the joint lower bound on the edge probability $p$ and the number of colors $kappa$ that ensures, with high probability, the presence of such a rainbow copy of $H$. Employing tools from random graph theory—including the first and second moment methods, coupling arguments, and refined counting of subgraphs under coloring constraints—we establish, for the first time, a unified joint threshold lower bound for rainbow embeddings of general spanning graphs $H$. Our main contribution lies in characterizing the interplay between $kappa$ and $p$: we show that if $p gg n^{-1/m_1(H)}$ and $kappa gg e(H)$, then a rainbow copy of $H$ appears asymptotically almost surely, where $m_1(H)$ denotes the $1$-density of $H$.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Social Networks and Social Media: Applications and emergent phenomena

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.

Problem

• Determine threshold for rainbow spanning subgraphs
• Establish lower bounds on edge probability p
• Find sufficient color count κ for existence

Innovation

• Edge-colored graphs with independent probabilities
• Rainbow spanning subgraph threshold analysis
• Lower bounds for probability and color count