Computing Chromatic Polynomials of Oriented Graphs

Let G = (V; A) be an antisymmetric directed graph. An oriented-coloring of G is deened as a mapping c from V to the set of colors f1;2;:::;g satisfying (i) 8 (x; y) 2 A; c(x) 6 = c(y) and (ii) 8 (x; y); (z; t) 2 A; c(x) = c(t) =) c(y) 6 = c(z). The oriented chromatic polynomial of G is then deened as the quantity ~ P (G;), standing for the number of oriented-colorings of G. We show in this paper how this polynomial can be computed and prove some properties of it. A) un graphe orient e antisym etrique. Une-coloration de G est une application c de V dans l'ensemble de couleurs f1;2;:::;g satisfaisant (i) 8 (x; y) 2 A; c(x) 6 = c(y) et (ii) 8 (x; y); (z; t) 2 A; c(x) = c(t) =) c(y) 6 = c(z). Le polyn^ ome chromatique de G est alors d eeni comme la quantit e ~ P(G;), repr esentant le nombre de-colorations de G. Nous montrons dans cet article comment ce polyn^ ome peut ^ etre calcul e et prouvons certaines propri et es.