On the chromatic numbers of Johnson type graphs

TL;DR

This paper investigates the chromatic numbers of Johnson type graphs with vertices in -1,0,1^n, revealing their growth rates for specific parameter choices, which are logarithmic or double logarithmic in n.

Contribution

It determines the asymptotic growth of chromatic numbers for certain Johnson type graphs, a problem previously not fully understood.

Findings

Chromatic number of J_(n,2,-1) grows logarithmically with n.

Chromatic number of J_(n,3,-1) grows logarithmically with n.

Chromatic number of J_(n,3,-2) grows double logarithmically with n.

Abstract

A Johnson type graph $J_{\pm}(n,k,t)$ is a graph whose vertex set consists of vectors from $\{-1,0,1\}^n$ of the length $\sqrt{k}$ and edges connect vertices with scalar product $t$. The paper determines the order of growth of the chromatic numbers of graphs $J_\pm(n,2,-1)$ and $J_\pm(n,3,-1)$ (logarithmic on $n$), and also $J_\pm(n,3,-2)$ (double logarithmic on $n$).