On directed local chromatic number, shift graphs, and Borsuk-like graphs

TL;DR

This paper explores the local chromatic number of shift graphs and related structures, revealing significant differences between directed and undirected cases and establishing bounds for certain topologically chromatic graphs.

Contribution

It demonstrates that the gap between directed and undirected local chromatic numbers can be arbitrarily large and determines the minimum directed local chromatic number for specific topologically t-chromatic graphs.

Findings

The local chromatic number of shift graphs is close to their chromatic number.

The gap between directed and undirected local chromatic numbers can be arbitrarily large.

Minimum directed local chromatic number for certain topologically t-chromatic graphs is approximately t/4+1.

Abstract

We investigate the local chromatic number of shift graphs and prove that it is close to their chromatic number. This implies that the gap between the directed local chromatic number of an oriented graph and the local chromatic number of the underlying undirected graph can be arbitrarily large. We also investigate the minimum possible directed local chromatic number of oriented versions of ``topologically t-chromatic'' graphs. We show that this minimum for large enough t-chromatic Schrijver graphs and t-chromatic generalized Mycielski graphs of appropriate parameters is the upper integer part of t/4+1.