Abstracts
for the Seminar

Spring 2022

**Speaker: **Saul Blanco, Indiana University

**Title: **Lengths of Cycles in Generalized Pancake Graphs

**Time:** 2:30 PM, Monday, May 9, 2022

**Place:** ZOOM

**Abstract:** 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 $m\geq3$, thus complementing the known results for $m=1,2$. This is joint work with Charles Buehrle.

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