Strong oriented chromatic number of planar graphs without short cycles

Let M be an additive abelian group. An M-strong-oriented coloring of an oriented graph G is a mapping ϕ from V (G) to M such that ϕ(u) 6 = ϕ(v) whenever −→uv is an arc in G and ϕ(v) − ϕ(u) 6 = −(ϕ(t) − ϕ(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 Nešetřil 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