Nordhaus-Guddam Type Relations of Three Graph Coloring Parameters

TL;DR

This paper investigates inequalities relating to three graph coloring parameters—2-proper, injective, and square chromatic numbers—extending Nordhaus-Guddam type relations to these parameters.

Contribution

It establishes new Nordhaus-Guddam type inequalities for three different graph coloring parameters, expanding theoretical understanding.

Findings

Derived inequalities for 2-proper chromatic number

Established bounds for injective chromatic number

Analyzed square chromatic number relations

Abstract

Let G be a simple graph. A coloring of vertices of G is called (i) a 2-proper coloring if vertices at distance 2 receive distinct colors; (ii) an injective coloring if vertices possessing a common neighbor receive distinct colors; (iii) a square coloring if vertices at distance at most 2 receive distinct colors. In this paper, we study inequalities of Nordhaus-Guddam type for the 2-proper chromatic number, the injective chromatic number, and the square chromatic number.