Finite Vertex-colored Ultrahomogeneous Oriented Graphs

TL;DR

This paper classifies all finite vertex-colored oriented ultrahomogeneous graphs with one asymmetric binary relation and multiple unary relations, providing a comprehensive understanding of their structure and construction methods.

Contribution

It offers a complete classification of finite vertex-colored oriented ultrahomogeneous graphs, including methods for constructing new examples and identifying exceptions.

Findings

Complete classification of such graphs.

Development of methods to combine directed graphs.

Identification of explicit exceptions.

Abstract

A relational structure R is ultrahomogeneous if every isomorphism of finite induced substructures of R extends to an automorphism of R. We classify the ultrahomogeneous finite binary relational structures with one asymmetric binary relation and arbitrarily many unary relations. In other words, we classify the finite vertex-colored oriented ultrahomogeneous graphs. The classification comprises several general methods with which directed graphs can be combined or extended to create new ultrahomogeneous graphs. Together with explicitly given exceptions, we obtain exactly all vertex-colored oriented ultrahomogeneous graphs this way. Our main technique is a technical tool that characterizes precisely under which conditions two binary relational structures with disjoint unary relations can be combined to form a larger ultrahomogeneous structure.