Outerplanar and planar oriented cliques

TL;DR

This paper investigates the structure and maximum size of outerplanar and planar oriented cliques, providing characterizations, confirming a conjecture about their size limit, and establishing bounds based on girth.

Contribution

It characterizes outerplanar oriented cliques via spanning subgraphs and confirms the maximum size of planar oriented cliques as 15, also providing bounds related to girth.

Findings

Outerplanar oriented cliques characterized by 11 specific graphs.

Maximum order of planar oriented cliques is 15, confirming a conjecture.

Provided bounds for planar oriented cliques with girth k for all k ≥ 4.

Abstract

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 $k \ge 4$.