Minimal non-extensible precolorings and implicit-relations

TL;DR

This paper investigates minimal non-extensible precolorings in graphs, revealing that only two vertices are needed for such precolorings regardless of the graph's chromatic number, and introduces the concept of implicit-relations.

Contribution

It develops new theory characterizing minimal non-extensible precolorings and introduces implicit-relations, providing insights into vertex coloring constraints.

Findings

Minimal non-extensible precolorings require only two precolored vertices.

The number of precolored vertices remains constant regardless of graph's chromatic number.

Introduces the concept of implicit-relations distinguishing implicit-edges and implicit-identities.

Abstract

In this paper I study a variant of the general vertex coloring problem called precoloring. Specifically, I study graph precolorings, by developing new theory, for characterizing the minimal non-extensible precolorings. It is interesting per se that, for graphs of arbitrarily large chromatic number, the minimal number of colored vertices, in a non-extensible precoloring, remains constant; only two vertices $u,v$ suffice. Here, the relation between such $u,v$ is called an implicit-relation, distinguishing two cases: (i) implicit-edges where $u,v$ are precolored with the same color and (ii) implicit-identities where $u,v$ are precolored distinct.