Nested colourings of graphs

TL;DR

This paper introduces the concept of nested colourings in graphs, showing how to compute the nested chromatic number efficiently and exploring its properties and bounds under various graph operations.

Contribution

It defines nested colourings, establishes polynomial-time computation of the nested chromatic number, and analyzes its behavior under different graph operations.

Findings

Nested chromatic number can be computed in polynomial time.

Provides bounds on the nested chromatic number based on graph properties.

Classifies possible nested chromatic numbers for graphs with fixed vertices and chromatic number.

Abstract

A proper vertex colouring of a graph is \emph{nested} if the vertices of each of its colour classes can be ordered by inclusion of their open neighbourhoods. Through a relation to partially ordered sets, we show that the nested chromatic number can be computed in polynomial time. Clearly, the nested chromatic number is an upper bound for the chromatic number of a graph. We develop multiple distinct bounds on the nested chromatic number using common properties of graphs. We also determine the behaviour of the nested chromatic number under several graph operations, including the direct, Cartesian, strong, and lexicographic product. Moreover, we classify precisely the possible nested chromatic numbers of graphs on a fixed number of vertices with a fixed chromatic number.