The Sum and Product of Chromatic Numbers of Graphs and their Line Graphs

TL;DR

This paper investigates bounds on the sum and product of chromatic numbers of graphs and their line graphs, providing new characterizations and extending Nordhaus-Gaddum-type results.

Contribution

It introduces a novel characterization of a class of graphs related to chromatic numbers and extends existing bounds to include line graphs.

Findings

Derived new bounds for chromatic numbers of line graphs

Provided a characterization of graphs related to chromatic number bounds

Extended Nordhaus-Gaddum-type results to line graphs

Abstract

A Nordhaus-Gaddum-type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. In this paper some variations are considered. First, recall their theorem, which gives bounds on the sum and the product of the chromatic number of a graph with that of its complement. In this paper, we provide a new characterization of the other class of graphs.