Cayley and Tutte polytopes

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 a 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 polytope. We prove that simplices in the triangulations correspond to labeled trees and forests. The heart of the proof is a direct bijection based on the neighbors-first search graph traversal algorithm.

Keywords

• polytopes
• volume
• cayley compositions
• tutte polynomial
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]