Outerplanar and planar oriented cliques
Ayan Nandy, Sagnik Sen, Eric Sopena

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 is the maximum order of a complete subgraph of and is a well-known lower bound for the chromatic number of . Every proper -coloring of may be viewed as a homomorphism (an edge-preserving vertex mapping) of to the complete graph of order . 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…
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