Integrated Neighborhood Colorings of Graphs

TL;DR

This paper introduces a generalized concept of unfriendly colorings called 'integrated' colorings for all n>1, proves their existence for all finite graphs, and explores applications to graph coloring and max-cut problems.

Contribution

It extends the concept of unfriendly colorings to a broader class called integrated colorings and establishes their universal existence for finite graphs.

Findings

Every finite graph has an integrated n-coloring for n>1.

Applications to various graph coloring problems.

Applications to some max-cut problems.

Abstract

The idea that those different from you are "unfriendly" is captured in the definition of unfriendly 2-colorings in graph theory in a paper by Aharoni, Milner and Prikry, where they prove that every finite graph has an unfriendly coloring. We give a more general definition for all n>1, that we call "integrated" rather than "unfriendly." Then we prove that every finite graph has an integrated n-coloring, n>1. We then give some applications to various graph coloring problems and to some max-cut problems.