On extremal properties of perfect 2-colorings

TL;DR

This paper explores the extremal properties of perfect 2-colorings in graphs, highlighting their connection to various bounds and introducing a new upper bound related to equitable partitions in regular graphs.

Contribution

It establishes a new upper bound for subsets with fixed average internal degree in amply regular graphs, linking equality cases to equitable partitions.

Findings

Perfect 2-colorings correspond to equality cases in several classical bounds.

The Expander Mixing Lemma characterizes perfect 2-colorings.

A new upper bound for subset size with fixed internal degree is proven.

Abstract

A coloring of vertices of a graph is called perfect if, for every vertex, the collection of colors of its neighbors depends only on its own color. The correspondent color partition of vertices is called equitable. We note that a number of bounds (Hoffman bound, Cheeger bound, Bierbrauer--Friedman bound and other) is only reached on perfect $2$-colorings. We show that the Expander Mixing Lemma is another example of an inequality that generates a perfect $2$-coloring. We prove a new upper bound for the size of $S\subset V(G)$ with the fixed average internal degree for an amply regular graph $G$. This bound is reached on the set $S$ if and only if $\{S, V(G)\setminus S\}$ is an equitable partition.