Automorphisms and regular embeddings of merged Johnson graphs

Abstract

The merged Johnson graph J( n, m) I is the union of the distance i graphs J( n, m) i of the Johnson graph J( n, m) for i∈ I, where ∅≠ I⊆{1,…, m} and 2≤ m≤ n/2. We find the automorphism groups of these graphs, and deduce that their only regular embedding in an orientable surface is the octahedral map on the sphere for J(4,2) 1, and that they have just six non-orientable regular embeddings. This yields classifications of the regular embeddings of the line graphs L( K n )= J( n,2) 1 of complete graphs, their complements L(K n) =J(n,2) 2 , and the odd graphs O m+1 = J(2 m+1, m) m .