Minimum Number of Monochromatic Subgraphs of a Random Graph

TL;DR

This paper investigates the minimal number of monochromatic subgraphs in a two-colored random graph, establishing convergence results and asymptotic formulas using graph theory and spin glass models.

Contribution

It introduces a new approach linking random graph subgraph appearance with hypergraph models and spin glass theory to analyze monochromatic subgraph minimization.

Findings

Minimum monochromatic subgraphs converge to a limit when expected copies are linear in graph size.

Asymptotic expression derived for the limit as subgraph size and expected copies diverge.

Connections established between random graph theory and spin glass models for this problem.

Abstract

We consider the problem of minimizing the number of monochromatic subgraphs of a random graph, when each node of the host graph is assigned one of the two colors. Using a recently discovered contiguity between appearance of strictly balanced subgraphs $F$ in a random graph, and random hypergraphs where copies of $F$ are generated independently, we show that the minimum value converges to a limit, when the expected number of copies of $F$ is linear in the number of nodes $|V|$. Furthermore, using the connections with mean field spin glass models, we obtain an asymptotic expression for this limit as the normalized expected number of copies of $F$ and the size of $F$ diverge to infinity.