Distance graphs having large chromatic numbers and not containing cliques or cycles of given size

TL;DR

This paper investigates the chromatic numbers of distance graphs in Euclidean space, establishing new lower bounds under constraints on clique sizes and girth, bridging combinatorial geometry and graph theory.

Contribution

It introduces novel lower bounds for the chromatic numbers of distance graphs with specific clique and girth restrictions, advancing understanding in combinatorial geometry.

Findings

New lower bounds for chromatic numbers of distance graphs

Results applicable to graphs with restricted clique sizes

Insights into the relationship between girth and chromatic number

Abstract

In this work, the classical Nelson -- Hadwiger problem is studied which lies on the edge of combinatorial geometry and graph theory. It concerns colorings of distance graphs in $ {\mathbb R}^n $, i.e., graphs such that their vertices are vectors and their edges are pairs of vectors at a distance from a given set of postive numbers apart. A series of new lower bounds are obtained for the chromatic numbers of such graphs with different restrictions on the clique numbers and the girths.