On the spanning trees of the hypercube and other products of graphs

TL;DR

This paper presents two combinatorial proofs for the number of spanning trees in the hypercube, introduces a general independence property, and extends results to hypercubes with diagonals and Cartesian products of complete graphs.

Contribution

It provides novel combinatorial proofs and extends the enumeration of spanning trees to more complex graph products, including hypercubes with diagonals and complete graph products.

Findings

Independence property of edge orientations in hypercubes

Enumeration formulas for hypercubes with diagonals

General formulas for Cartesian products of complete graphs

Abstract

We give two combinatorial proofs of an elegant product formula for the number of spanning trees of the $n$-dimensional hypercube. The first proof is based on the assertion that if one chooses a uniformly random rooted spanning tree of the hypercube and orient each edge from parent to child, then the parallel edges of the hypercube get orientations which are independent of one another. This independence property actually holds in a more general context and has intriguing consequences. The second proof uses some "killing involutions" in order to identify the factors in the product formula. It leads to an enumerative formula for the spanning trees of the $n$-dimensional hypercube augmented with diagonals edges, counted according to the number of edges of each type. We also discuss more general formulas, obtained using a matrix-tree approach, for the number of spanning trees of the Cartesian product of complete graphs.