Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Outerplanar and planar oriented cliques   The clique number of an undirected graph $G$ is the maximum order of a
complete subgraph of $G$ and is a well-known lower bound for the chromatic
number 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 of
length at most 2), we get a natural notion of (oriented) colorings and oriented
chromatic number of oriented graphs. An oriented clique is then an oriented
graph whose number of vertices and oriented chromatic number coincide. However,
the structure of oriented cliques is much less understood than in the
undirected case.
  In this paper, we study the structure of outerplanar and planar oriented
cliques. We first provide a list of 11 graphs and prove that an outerplanar
graph can be oriented as an oriented clique if and only if it contains one of
these graphs as a spanning subgraph. Klostermeyer and MacGillivray conjectured
that the order of a planar oriented clique is at most 15, which was later
proved 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 any
planar oriented clique on 15 vertices must contain a particular oriented graph
as a spanning subgraph, thus reproving the above conjecture. We also provide
tight upper bounds for the order of planar oriented cliques of girth $k$ for
all $k \ge 4$.

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The categories of this paper are: mtrl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: nlin
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The categories of this paper are: q-bio
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-lat
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The categories of this paper are: gr-qc
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: funct-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: math-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-th