Chromatic Polynomials of Oriented 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.

Chromatic polynomials of 2-edge coloured graphs

“…By way of homomorphism, one can define, for each (m, n) = (0, 0), colouring for (m, n)-mixed graphs that generalizes graph colouring, oriented graph colouring and 2-edge-coloured graph colouring. As our results in Section 2 closely mirror those in [7] for oriented graphs we expect that the results in Section 2 in fact special cases of a more general result for the, to be defined, chromatic polynomial of an (m, n)-mixed graph. Showing such a result would require successfully generalizing the notions of obstructing arcs/edge as well as the notions of 2-dipath and bichromatic 2-path.…”

“…The results and methods in Section 2 closely mirror those for the oriented chromatic polynomial in [7]. Such a phenomenon has been observed in the study of the chromatic number oriented graphs and 2-edge-coloured graphs.…”

“…In [7] the authors fully characterize those oriented graphs that admit a chromatically invariant orientation. Theorem 3.8 presents a partial analogue for 2-edge-coloured graphs.…”

An Analogue of Quasi-Transitivity for Edge-Coloured Graphs

“…Such a phenomenon also occurs with the study of chromatic polynomials of oriented and 2-edge-coloured graphs. In [5], Cox and Duffy fully classified those oriented graphs whose oriented chromatic polynomial was identical to the chromatic polynomials of the underlying simple graph as quasi-transitive orientations of cointerval graphs, notably a class of graphs for which the chromatic polynomial can be computed in polynomial time [8]. In the sequel for 2-edge-coloured graphs, Beaton, Cox, Duffy and Zolkavich [3] obtain only a partial classification of 2-edge-coloured graphs whose 2-edge-coloured chromatic polynomial is identical to the chromatic polynomial of the underlying simple graph.…”

A Homomorphic Polynomial for Oriented Graphs

“…He also noticed that this function lacks two very important properties, namely, the alternating signs of the coefficients property (present in chromatic polynomials) and the unimodal behavior of the absolute values of the coefficients property (present in chromatic polynomial, proved recently by Huh and Katz [9] through positively settling a long standing conjecture). Recently, Cox and Duffy [5] have made considerable progress on this topic by establishing several interesting properties of the chromatic polynomial. Looking back at the two decades of research on oriented coloring, it can be noted that, homomorphism, alongside coloring, is also an important aspect of oriented coloring.…”