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The clique number of an undirected graph $G$ is the maximum order of acomplete subgraph of $G$ and is a well-known lower bound for the chromaticnumber of $G$. Every proper $k$-coloring of $G$ may be viewed as a homomorphism(an edge-preserving vertex mapping) of $G$ to the complete graph of order $k$.By considering homomorphisms of oriented graphs (digraphs without cycles oflength at most 2), we get a natural notion of (oriented) colorings and orientedchromatic number of oriented graphs. An oriented clique is then an orientedgraph whose number of vertices and oriented chromatic number coincide. However,the structure of oriented cliques is much less understood than in theundirected case.  In this paper, we study the structure of outerplanar and planar orientedcliques. We first provide a list of 11 graphs and prove that an outerplanargraph can be oriented as an oriented clique if and only if it contains one ofthese graphs as a spanning subgraph. Klostermeyer and MacGillivray conjecturedthat the order of a planar oriented clique is at most 15, which was laterproved by Sen [S. Sen. Maximum Order of a Planar Oclique Is 15. Proc.IWOCA'2012. {\em Lecture Notes Comput. Sci.} 7643:130--142]. We show that anyplanar oriented clique on 15 vertices must contain a particular oriented graphas a spanning subgraph, thus reproving the above conjecture. We also providetight upper bounds for the order of planar oriented cliques of girth $k$ forall $k \ge 4$.

Outerplanar and planar oriented cliques