Some New Results on Proper Colouring of Edge-set Graphs

TL;DR

This paper investigates the proper coloring of edge-set graphs, focusing on paths, and establishes their chromatic number and perfection, contributing foundational results to graph coloring theory.

Contribution

It provides the first derivation of the chromatic number for edge-set graphs of paths and proves these graphs are perfect, advancing understanding in graph coloring.

Findings

Chromatic number of edge-set graphs of paths derived

Edge-set graphs of paths are perfect graphs

Foundational results for proper coloring of edge-set graphs

Abstract

In this paper, we present a foundation study for proper colouring of edge-set graphs. The authors consider that a detailed study of the colouring of edge-set graphs corresponding to the family of paths is best suitable for such foundation study. The main result is deriving the chromatic number of the edge-set graph of a path, $P_{n+1}$, $n \geq 1$. It is also shown that edge-set graphs for paths are perfect graphs.