Let M be an additive abelian group. An M-strong-oriented coloring of an oriented graph G is a mapping f from V(G) to M such that f(u) <> j(v) whenever uv is an arc in G and f(v)−f(u) <> −(f(t)−f(z)) whenever uv and zt are two arcs in G. The strong oriented chromatic number of an oriented graph is the minimal order of a group M such that G has an M-strong-oriented coloring. This notion was introduced by Nesetril and Raspaud [Ann. Inst. Fourier, 49(3):1037-1056, 1999]. We prove that the strong oriented chromatic number of oriented planar graphs without cycles of lengths 4 to 12 (resp. 4 or 6) is at most 7 (resp. 19). Moreover, for all i ≥ 4, we construct outerplanar graphs without cycles of lengths 4 to i whose oriented chromatic number is 7.
Discrete Mathematics & Theoretical Computer Science