Isomorphisms of Cayley multigraphs of degree 4 on finite Abelian groups (Q1185868)

From the standard definition of the Cayley graph \(\Gamma_ H(G)\) of a group \(G\) with respect to set \(H\), the authors consider the following generalizations: a Cayley multigraph is obtained if elements of \(H\) are permitted arbitrary positive multiplicities; a directed Cayley graph is obtained if it is not required that \(H=H^{-1}\); \(G\) need not be connected if \(H\) need not generate \(G\). Two of these objects \(\Gamma_ H(G)\) and \(\Gamma_{H'}(G')\) are Adám isomorphic if there exists a group-isomorphism \(\tau:G\to G'\) such that \(\tau[H]=H'\) while preserving multiplicities. It is shown that any two finite isomorphic (connected) Cayley multigraphs of valence 4 coming from Abelian groups are Adám isomorphic (with one pair of exceptions). From this one proves Adám's circulant graph conjecture for the case of valence 4.