Lengths of Cycles in Generalized Pancake Graphs
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In this paper, we consider the lengths of cycles that can be embedded on the edges of the generalized pancake graph which is the Cayley graph of the generalized symmetric group, the wreath product of the cyclic group C_m and the symmetric group, generated by prefix reversals. In the cases when the cyclic group has one or two elements the graphs are the pancake graphs and burnt pancake graphs, respectively. We prove that when the cyclic group has three elements the underlying, undirected graph of the generalized pancake graph is pancyclic, thus resembling a similar property of the pancake graphs and the burnt pancake graphs. Moreover, when the cyclic group has four elements, the resulting undirected graphs will have all the even-length cycles. We utilize these results as base cases and show that if m>2 is even, the corresponding undirected pancake graph has all cycles of even length starting from its girth to a Hamiltonian cycle. Moreover, when m is odd, the corresponding undirected pancake graph has cycles of all lengths starting from its girth to a Hamiltonian cycle. We furthermore show that the girth of the undirected generalized pancake graphs is min{m,8} if m≥3, thus complementing the known results for m=1,2.
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