Classification of Finite Highly Regular Vertex-Coloured Graphs

Abstract

A coloured graph is k-ultrahomogeneous if every isomorphism between two induced subgraphs of order at most k extends to an automorphism. A coloured graph is k-tuple regular if the number of vertices adjacent to every vertex in a set S of order at most k depends only on the isomorphism type of the subgraph induced by S.We classify the finite vertex-coloured k-ultrahomogeneous graphs and the finite vertexcoloured k-tuple regular graphs for k ≥ 4. Our theorem in particular classifies finite vertexcoloured ultrahomogeneous graphs, where ultrahomogeneous means the graph is simultaneously k-ultrahomogeneous for all k ∈ N. * The research leading to these results has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (EngageS: grant agreement No. 820148).1 Some authors use the term homogeneous instead. We use the term ultrahomogeneous to highlight that every and not just some isomorphism between isomorphic substructures must extend to an automorphism.