Extremal regular graphs of given chromatic number

TL;DR

This paper investigates the smallest possible regular graphs with a given chromatic number, identifying specific extremal graphs like Turán graphs, antihole graphs, and certain Cayley graphs, and explores their properties and constructions.

Contribution

It characterizes extremal regular graphs with fixed chromatic number, including new constructions from Turán graphs and analysis of Cayley graph cases.

Findings

Turán graphs $T_{ak,k}$ are extremal for given parameters

Antihole graphs are extremal in this context

Certain Cayley graphs are also extremal and can be constructed from Turán graphs

Abstract

We define an extremal $(r|\chi)$-graph as an $r$-regular graph with chromatic number $\chi$ of minimum order. We show that the Tur{\' a}n graphs $T_{ak,k}$, the antihole graphs and the graphs $K_k\times K_2$ are extremal in this sense. We also study extremal Cayley $(r|\chi)$-graphs and we exhibit several $(r|\chi)$-graph constructions arising from Tur{\' a}n graphs.