Incidence coloring of Regular graphs and Complement graphs

TL;DR

This paper explores the conditions under which regular graphs can be incidence colored with a specific number of colors and establishes optimal inequalities relating to their incidence chromatic number.

Contribution

It provides necessary and sufficient conditions for r-regular graphs to be (r+1)-incidence colorable and determines the optimal Nordhaus-Gaddum inequality for the incidence chromatic number.

Findings

Characterization of r-regular graphs that are (r+1)-incidence colorable

Optimal Nordhaus-Gaddum inequality for incidence chromatic number

Relation between domination number and incidence chromatic number

Abstract

Using a relation between domination number and incidence chromatic number, we obtain necessary and sufficient conditions for $r$-regular graphs to be $(r+1)$-incidence colorable. Also, we determine the optimal Nordhaus-Gaddum inequality for the incidence chromatic number.