2024/04/14 by Shirin Alimirzaei, Dave Witte Morris, Alimirzaei, Shirin +1
Biochemistry, Genetics and Molecular Biology · Mathematics · Social Sciences · #Abelian group #Automorphism #Automorphism group #Cayley graph #Combinatorics #Discrete mathematics #Graph #Line graph #Mathematics #Migration, Ethnicity, and Economy #Vertex (graph theory) #Vertex-transitive graph #Voltage graph #math.CO #melanin and skin pigmentation
paper · pdf · doi:10.48550/arxiv.2404.09367
openalex publication_date 2024/04/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let G be a (finite or infinite) group, and let KG = Cay(G;G \smallsetminus \1\ ) be the complete graph with vertex set G, considered as a Cayley graph of G. Being a Cayley graph, it has a natural edge-colouring by sets of the form \s, s-1\ for s ∈ G. We prove that every colour-permuting automorphism of KG is an affine map, unless G ≅ Q8 × B, where Q8 is the quaternion group of order 8, and B is an abelian group, such that b2 is trivial for all b ∈ B. We also prove (without any restriction on G) that every colour-permuting automorphism of KG is the composition of a group automorphism and a colour-preserving graph automorphism. This was conjectured by D. P. Byrne, M. J. Donner, and T. Q. Sibley in 2013.