Pancyclicity and Cayley graphs on generalized dihedral groups (Q286758)

A generalized dihedral group \(D_H\) is generated by an abelian group \(H\) and an involution \(g\in D_H\setminus H\) such that \(ghg=h^{-1}\) for all \(h\in H\). Let \(X=\mathrm{Cay}(D_H,S)\) denote the Cayley graph of \(D_H\) with connection set \(S\), where \(|S|\geq3\). Let \(\gamma\) (resp., \(\gamma_o\)) denote the length of a shortest cycle (resp., odd cycle) in \(X\). Suppose that \(S\cap H\neq\emptyset\). Then, the following two results hold: 1) If \(X\) is bipartite, then \(X\) is even edge-pancyclic, that is, every edge lies on some cycle of each even length from 4 through \(2\lfloor|D_H|/2\rfloor\). 2) If \(X\) is not bipartite, then \(X\) is weakly odd vertex-pancyclic, that is, every vertex lies on some cycle of each odd length \(\geq\gamma_o\) and not exceeding the circumference of \(X\). This extends a result for abelian groups by \textit{B. Alspach} et al. [J. Graph Theory 74, No. 3--4, 260--274 (2013; Zbl 1276.05058)].