A universal dichotomy for concentration in randomly colored graphs

TL;DR

This paper establishes a universal dichotomy in the concentration behavior of subgraph sizes in randomly colored graphs, depending on the Euclidean norm of their degree sequences.

Contribution

It introduces a new dichotomy framework that characterizes concentration regimes in randomly colored graphs based on degree sequence norms.

Findings

For $oldsymbol{ ext{zeta}}=o(1)$, subgraph sizes concentrate around expected values.

For $oldsymbol{ ext{zeta}}= heta(1)$, concentration depends on color balance.

In the balanced case, the total number of monochromatic edges concentrates.

Abstract

Let $\zeta$ be Euclidean norm of the degree sequence of a graph normalized by the graph size. We prove that when the vertices of a graph are randomly colored with $s$ colors such that the fraction of vertices in each color class is bounded away from zero, only two asymptotic regimes emerge. If $\zeta=o(1)$, then the sizes of the subgraphs induced by the color classes concentrate around their expected values. If $\zeta=\Theta(1)$, then concentration depends on the color balance: for colorings with persisting imbalance, the total number $M$ of monochromatic edges stays bounded away from its mean with positive probability; otherwise, for vanishing imbalance, $M$ still concentrates. The same dichotomy holds for a broad class of randomly colored random graphs.