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

Abstract

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 $\overrightarrow{uv}$ and $\overrightarrow{zt}$ of G, we have $\varphi (v)-\varphi (u)\ne -(\varphi (t)-\varphi (z))$. The strong oriented chromatic number ${\chi }_s(G)$ 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?