Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On extremal properties of perfect 2-colorings

Vladimir N. Potapov

TL;DR

The paper investigates extremal properties of perfect $2$-colorings in regular and amply regular graphs, highlighting that many tight bounds are realized only by such colorings. It shows the Expander Mixing Lemma yields equality precisely for perfect $2$-colorings, linking spectral structure to equitable partitions. A new upper bound for the size of a subset with fixed average internal degree in amply regular graphs is derived, with equality characterized by perfect colorings, and the results imply stronger bounds than Hoffman in some cases. The work also connects these extremal properties to bounds on Boolean function sensitivity on Hamming graphs, illustrating a broad link between spectral graph theory, extremal combinatorics, and design theory.

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.

On extremal properties of perfect 2-colorings

TL;DR

The paper investigates extremal properties of perfect -colorings in regular and amply regular graphs, highlighting that many tight bounds are realized only by such colorings. It shows the Expander Mixing Lemma yields equality precisely for perfect -colorings, linking spectral structure to equitable partitions. A new upper bound for the size of a subset with fixed average internal degree in amply regular graphs is derived, with equality characterized by perfect colorings, and the results imply stronger bounds than Hoffman in some cases. The work also connects these extremal properties to bounds on Boolean function sensitivity on Hamming graphs, illustrating a broad link between spectral graph theory, extremal combinatorics, and design theory.

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 -colorings. We show that the Expander Mixing Lemma is another example of an inequality that generates a perfect -coloring. We prove a new upper bound for the size of with the fixed average internal degree for an amply regular graph . This bound is reached on the set if and only if is an equitable partition.
Paper Structure (4 sections, 15 theorems, 29 equations)

This paper contains 4 sections, 15 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Expander Mixing Lemma and its corollaries
  4. Perfect $2$-colorings of amply regular graphs

Key Result

Proposition 1

A two valued function $f:V\rightarrow \mathbb{R}$ is a perfect $2$-coloring of a regular connected graph $G(V,E)$ if and only if there exists a constant $\gamma$ such that $f-\gamma{\mathbf 1}_V$ is an eigenfunction of $G$ with eigenvalue $\lambda$, where $\lambda$ is an eigenvalue of the correspond

Theorems & Definitions (15)

  • Proposition 1: FdF2
  • Proposition 2
  • Proposition 3
  • Theorem 1: PotAv
  • Corollary 1
  • Lemma 1: Expander Mixing Lemma
  • Corollary 2: Golubev
  • Corollary 3
  • Corollary 4: Pot12, Theorem 1
  • Corollary 5
  • ...and 5 more