Chromatic Polynomials of Oriented Graphs

TL;DR

This paper classifies oriented graphs whose chromatic polynomial matches that of their underlying simple graph, characterizes them as quasi-transitive oriented co-interval graphs, and explores their polynomial roots.

Contribution

It fully classifies such oriented graphs, providing a polynomial-time identification method and analyzing the roots of their chromatic polynomials.

Findings

Identified and constructed all oriented graphs with matching chromatic polynomials to their underlying graphs.

Proved that these graphs are exactly the quasi-transitive oriented co-interval graphs.

Discovered that some oriented graphs have chromatic polynomial roots not realizable by simple graphs.

Abstract

The oriented chromatic polynomial of a oriented graph outputs the number of oriented $k$-colourings for any input $k$. We fully classify those oriented graphs for which the oriented graph has the same chromatic polynomial as the underlying simple graph, closing an open problem posed by Sopena. We find that such oriented graphs can be both identified and constructed in polynomial time as they are exactly the family of quasi-transitive oriented co-interval graphs. We study the analytic properties of this polynomial and show that there exist oriented graphs which have chromatic polynomials have roots, including negative real roots, that cannot be realized as the root of any chromatic polynomial of a simple graph.