On a $\vec{C}_4$-ultrahomogeneous oriented graph

TL;DR

This paper introduces a new strongly connected oriented graph with specific symmetry properties, constructed by modifying the Coxeter graph using combinatorial structures from the Fano plane, and demonstrates its unique features.

Contribution

It adapts the concept of ultrahomogeneity to digraphs and constructs a novel $oldsymbol{C}_4$-ultrahomogeneous oriented graph with specific symmetry and cycle properties.

Findings

The graph has 168 vertices and 126 arc-disjoint 4-cycles.

It is strongly connected with regular indegree and outdegree 3.

The graph contains no circuits of lengths 2 or 3.

Abstract

The notion of a $\mathcal C$-ultrahomogeneous graph, due to Isaksen et al., is adapted for digraphs, and subsequently a strongly connected $\vec{C}_4$-ultrahomogeneous oriented graph on 168 vertices and 126 pairwise arc-disjoint 4-cycles is presented, with regular indegree and outdegree 3 and no circuits of lengths 2 and 3, by altering a definition of the Coxeter graph via pencils of ordered lines of the Fano plane in which pencils are replaced by ordered pencils.