Article ID Journal Published Year Pages File Type
6425816 Advances in Mathematics 2013 33 Pages PDF
Abstract

Cayley polytopes were defined recently as convex hulls of Cayley compositions introduced by Cayley in 1857. In this paper we resolve Braun's conjecture, which expresses the volume of Cayley polytopes in terms of the number of connected graphs. We extend this result to two one-variable deformations of Cayley polytopes (which we call t-Cayley and t-Gayley polytopes), and to the most general two-variable deformations, which we call Tutte polytopes. The volume of the latter is given via an evaluation of the Tutte polynomial of the complete graph.Our approach is based on an explicit triangulation of the Cayley and Tutte polytopes. We prove that simplices in the triangulations correspond to labeled trees. The heart of the proof is a direct bijection based on the neighbors-first search graph traversal algorithm.

Related Topics
Physical Sciences and Engineering Mathematics Mathematics (General)
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