Cayley properties of merged Johnson graphs

TL;DR

This paper characterizes when merged Johnson graphs are Cayley graphs or have specific automorphism properties, extending previous results and identifying the limited cases where these symmetries occur.

Contribution

It provides a complete characterization of Cayley and nearly Cayley merged Johnson graphs, expanding understanding of their automorphism groups.

Findings

Identifies conditions for merged Johnson graphs to be Cayley graphs.

Classifies merged Johnson graphs with transitive automorphism groups and small vertex stabilizers.

Shows that only few merged Johnson graphs are Cayley or nearly Cayley graphs.

Abstract

Extending earlier results of Godsil and of Dobson and Malnic on Johnson graphs, we characterise those merged Johnson graphs $J=J(n,k)_I$ which are Cayley graphs, that is, which are connected and have a group of automorphisms acting regularly on the vertices. We also characterise the merged Johnson graphs which are not Cayley graphs but which have a transitive group of automorphisms with vertex-stabilisers of order $2$. Even though these merged Johnson graphs are all vertex-transitive, we show that only relatively few of them are Cayley graphs or have a transitive group of automorphisms with vertex-stabilisers of order $2$.