We define the excess degree e(P) of a d-polytope P as 2f_1-df_0, where f_0 and f_1 denote the number of vertices and edges of the polytope, respectively. We first prove that the excess degree of a d-polytope does not take every natural number: the smallest values are 0 and d-2, with the value d-1 only occurring when d=3 or 5. On the other hand, if d is even, the excess degree takes every even natural number from dsqrt{d} onwards; while, if d is odd, the excess degree takes every natural number from dsqrt{2d} onwards. We also revisit some known results and conjectures on the hamiltonicity of graphs of polytopes.
About the speaker:
In 2002 Dr Pineda-Villavicencio completed a Bachelor degree in Computer Science with first class honors at the University of Oriente, Santiago de Cuba, Cuba. In December 2009 he graduated in a Doctor of Philosophy from Federation University Australia. Between 2002 and 2008 Dr Pineda-Villavicencio worked as a lecturer in mathematics at the University of Information Sciences (Cuba) and the University of Oriente (Cuba). Between January 2009 and February 2013 he was a postdoctoral fellow under Prof John Yearwood and a lecturer in Mathematics at Federation University Australia. Guillermo spent the period between February and July 2013 at the Ben Gurion University of the Negev (Israel), under Dr Eran Nevo and Prof Mikhail Klin. Since July 2013 he is a lecturer in Mathematics at FedUni.
Dr Pineda-Villavicencio’s research deals with the theory and application of combinatorics. In particular, he pursues three main research streams: analysis of large networks, combinatorics of convex polytopes, and applications of combinatorics to health informatics.
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