Relations between the local chromatic number and its directed version

TL;DR

This paper investigates the relationship between local chromatic numbers in undirected and directed graphs, revealing differences in their fractional versions and establishing bounds on their ratios.

Contribution

It demonstrates the existence of graphs where directed local chromatic numbers are strictly less than undirected ones, and determines the supremum ratio for fractional parameters as e.

Findings

Existence of graphs with lower directed local chromatic number

Fractional parameters always align for some orientations

Supremum ratio of fractional parameters is e

Abstract

The local chromatic number is a coloring parameter defined as the minimum number of colors that should appear in the most colorful closed neighborhood of a vertex under any proper coloring of the graph. Its directed version is the same when we consider only outneighborhoods in a directed graph. For digraphs with all arcs being present in both directions the two values are obviously equal. Here we consider oriented graphs. We show the existence of a graph where the directed local chromatic number of all oriented versions of the graph is strictly less than the local chromatic number of the underlying undirected graph. We show that for fractional versions the analogous problem has a different answer: there always exists an orientation for which the directed and undirected values coincide. We also determine the supremum of the possible ratios of these fractional parameters, which turns out to be e, the basis of the natural logarithm.