On multichromatic numbers of widely colorable graphs

TL;DR

This paper determines the exact multichromatic number for widely colorable graphs, answering a specific open question and exploring asymptotic behaviors related to fractional chromatic numbers.

Contribution

It proves that the $r^{th}$ multichromatic number of $W(s,t)$ equals $t+2(s-1)$ for all $s$, resolving Tardif's question.

Findings

The $r^{th}$ multichromatic number $ ext{chi}_r(W(s,t))$ equals $t+2(s-1)$ for all $s$.

For large $r$, equality does not hold, indicating limits of the multichromatic number.

The fractional chromatic number of $W(s,t)$ tends to infinity as $t$ increases.

Abstract

A coloring is called $s$-wide if no walk of length $2s-1$ connects vertices of the same color. A graph is $s$-widely colorable with $t$ colors if and only if it admits a homomorphism into a universal graph $W(s,t)$. Tardif observed that the value of the $r^{\rm th}$ multichromatic number $\chi_r(W(s,t))$ of these graphs is at least $t+2(r-1)$ and equality holds for $r=s=2$. He asked whether there is equality also for $r=s=3$. We show that $\chi_s(W(s,t))=t+2(s-1)$ for all $s$ thereby answering Tardif's question. We observe that for large $r$ (with respect to $s$ and $t$ fixed) we cannot have equality and that for $s$ fixed and $t$ going to infinity the fractional chromatic number of $W(s,t)$ also tends to infinity. The latter is a simple consequence of another result of Tardif on the fractional chromatic number of generalized Mycielski graphs.