Strong oriented chromatic number of planar graphs without cycles of specific lengths

A strong oriented k-coloring of an oriented graph G is a homomorphism ϕ from G to H having k vertices labelled by the k elements of an abelian additive group M, such that for any pairs of arcs u v → and z t → of G, we have ϕ ( v ) − ϕ ( u ) ≠ − ( ϕ ( t ) − ϕ ( z ) ) . The strong oriented chromatic number χ s ( G ) is the smallest k such that G admits a strong oriented k-coloring. In this paper, we consider the following problem: Let i ⩾ 4 be an integer. Let G be an oriented planar graph without cycles of lengths 4 to i. What is the strong oriented chromatic number of G?