Triangulations of Cayley and Tutte polytopes

TL;DR

This paper proves Braun's conjecture relating Cayley polytope volumes to connected graphs, extends it to deformations called t-Cayley, t-Gayley, and Tutte polytopes, and provides explicit triangulations linking simplices to labeled trees.

Contribution

It resolves Braun's conjecture, introduces new deformations of Cayley polytopes, and constructs explicit triangulations connecting geometric and combinatorial structures.

Findings

Volume of Cayley polytopes equals the number of connected graphs.

Triangulations correspond to labeled trees via a neighbor-first search.

Volumes of deformed polytopes are expressed through evaluations of the Tutte polynomial.

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 polytope. 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.