An improved bound on the chromatic number of the Pancake graphs

TL;DR

This paper presents an improved bound on the chromatic number of Pancake graphs using subadditivity, explores equitable colorings, and conjectures equality between chromatic and equitable chromatic numbers for all n.

Contribution

It introduces a new bound on the chromatic number of Pancake graphs and investigates equitable colorings, including a specific equitable $(n-1)$-coloring and optimal small n colorings.

Findings

An improved bound on the chromatic number of Pancake graphs.

An equitable $(n-1)$-coloring based on efficient dominating sets.

Conjecture that chromatic and equitable chromatic numbers are equal for all n.

Abstract

In this paper an improved bound on the chromatic number of the Pancake graph $P_n, n\geqslant 2$, is presented. The bound is obtained using a subadditivity property of the chromatic number of the Pancake graph. We also investigate an equitable coloring of $P_n$. An equitable $(n-1)$-coloring based on efficient dominating sets is given and optimal equitable $4$-colorings are considered for small $n$. It is conjectured that the chromatic number of $P_n$ coincides with its equitable chromatic number for any $n\geqslant 2$.