On the oriented chromatic number of Halin graphs

An oriented k-coloring of an oriented graph G is a mapping c : V (G) → {1, 2, . . . , k} such that (i) if xy ∈ E(G) then c(x) 6= c(y) and (ii) if xy, zt ∈ E(G) then c(x) = c(t) =⇒ c(y) 6= c(z). The oriented chromatic number ~ χ(G) of an oriented graph G is defined as the smallest k such that G admits an oriented k-coloring. We prove in this paper that every Halin graph has oriented chromatic number at most 9, improving a previous bound proposed by Vignal. AMS Subject Classification: 05C15.