On the oriented chromatic number of Halin graphs

Abstract

An oriented k-coloring of an oriented graph G is a mapping c : V ( G ) → { 1 , 2 , … , k } such that (i) if x y ∈ E ( G ) then c ( x ) ≠ c ( y ) and (ii) if x y , z t ∈ E ( G ) then c ( x ) = c ( t ) ⇒ c ( y ) ≠ 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.