Which Haar graphs are Cayley graphs?

2015/05/06 by Estélyi, István, Pisanski, Tomaž · 1 citation
#05C25 #05E10 #20B25 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1505.01475

Abstract

For a finite group G and subset S of G, the Haar graph H(G,S) is a bipartite regular graph, defined as a regular G-cover of a dipole with |S| parallel arcs labelled by elements of S. If G is an abelian group, then H(G,S) is well-known to be a Cayley graph; however, there are examples of non-abelian groups G and subsets S when this is not the case. In this paper we address the problem of classifying finite non-abelian groups G with the property that every Haar graph H(G,S) is a Cayley graph. An equivalent condition for H(G,S) to be a Cayley graph of a group containing G is derived in terms of G, S and Aut G. It is also shown that the dihedral groups, which are solutions to the above problem, are ℤ₂²,D₃,D₄ and D₅.

Cited by

• On kernel isomorphisms of m-Cayley digraphs and finite 2PCI-groups

Related

• On groups all of whose Haar graphs are Cayley graphs
• Existence of non-Cayley Haar graphs
• Two families of graphs that are Cayley on nonisomorphic groups
• Cayley graphs on non-isomorphic groups
• On the power graphs which are Cayley graphs of some groups