Finite Vertex-colored Ultrahomogeneous Oriented Graphs

TL;DR

This work classifies all finite vertex-colored oriented ultrahomogeneous graphs, i.e., finite binary relations with one asymmetric edge relation and arbitrary unary colorings, up to color changes and bichromatic symmetrization. The authors develop a general extension theorem that gives five necessary and sufficient conditions to glue two ultrahomogeneous sides into a larger ultrahomogeneous structure, and they introduce a flexible blow-up framework that preserves ultrahomogeneity in a controlled way. By combining these tools with Lachlan’s classification of monochromatic ultrahomogeneous digraphs, they obtain an explicit, finite list of building blocks and blow-up patterns from which every finite vertex-colored oriented ultrahomogeneous graph can be constructed. The results advance the understanding of highly symmetric vertex-colored relational structures and provide a method that generalizes to broader binary-relational settings, with potential algorithmic benefits for symmetry detection and exploitation.

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.

Figures (5)

• Figure 1: Example of an ultrahomogeneous oriented vertex colored graph (no blow-ups).
• Figure 2: The ultrahomogeneous oriented (monochromatic) graphs. For each of the three infinite families $\{E_n\colon n \in \mathbb{N}_{\geq 1} \}$, $\{E_n \cdot \mathop{\overrightarrow{C_3}}\colon n \in \mathbb{N}_{\geq 1} \}$, and $\{\mathop{\overrightarrow{C_3}} \cdot E_n\colon n \in \mathbb{N}_{\geq 1}\}$ we show one example. An arrow from one gray circle to another indicates all arcs from vertices in the first circle to vertices in the second are present.
• Figure 3: Reflecting the arcs matching a red $E_2$ to a blue $E_2$ via bichromatic symmetrizations and inverse bichromatic symmetrizations.
• Figure 4: Up to color changes and bichromatic symmetrization, these are the ultrahomogeneous bichromatic oriented graphs containing $\mathop{\overrightarrow{C_4}}$ as a color class.
• Figure 5: Up to equivalence all ultrahomogeneous oriented bichromatic graphs can be obtained from blow-ups of graphs of the above form.

Theorems & Definitions (47)

• Remark 2.1
• Theorem 2.2: lachlan82-finite-homogeneous-simple-digraphs
• Definition 3.1: Ultrahomogeneous system of partitions
• Definition 3.2: Easygoing CCD with respect to a block system
• Lemma 3.3
• proof
• Theorem 3.4: General extension theorem
• proof
• Theorem 3.5: Minimal extension theorem
• proof