Characterisation and classification of signatures of spanning trees of the $n$-cube

The signature of a spanning tree $T$ of the $n$-cube $Q_n$ is the $n$-tuple $\mathrm{s}\mathrm{i}\mathrm{g}(T)=(a_1,a_2,\dots ,a_n)$ such that $a_i$ is the number of edges of $T$ in the $i$th direction. We characterise the $n$-tuples that can occur as the signature of a spanning tree, and classify a signature $\mathcal{S}$ as reducible or irreducible according to whether or not there is a proper nonempty subset $R$ of $[n]$ such that restricting $\mathcal{S}$ to the indices in $R$ gives a signature of $Q_{\mathrm{\mid }R\mathrm{\mid }}$. If so, we say moreover that $\mathcal{S}$ and $T$ reduce over $R$. We show that reducibility places strict structural constraints on $T$. In particular, if $T$ reduces over a set of size $r$ then $T$ decomposes as a sum of $2^r$ spanning trees of $Q_{n-r}$, together with a spanning tree of a contraction of $Q_n$ with underlying simple graph $Q_r$. Moreover, this decomposition is realised by an isomorphism of edge slide graphs, where the edge slide graph of $Q_n$ is the graph $\mathcal{E}(Q_n)$ on the spanning trees of $Q_n$, with an edge between two trees if and only if they are related by an edge slide. An edge slide is an operation on spanning trees of the $n$-cube given by ``sliding’’ an edge of a spanning tree across a $2$-dimensional face of the cube to get a second spanning tree. The signature of a spanning tree is invariant under edge slides, so the subgraph $\mathcal{E}(\mathcal{S})$ of $\mathcal{E}(Q_n)$ induced by the trees with signature $\mathcal{S}$ is a union of one or more connected components of $\mathcal{E}(Q_n)$. Reducible signatures may be further divided into strictly reducible and quasi-irreducible signatures, and as an application of our results we show that $\mathcal{E}(\mathcal{S})$ is disconnected if $\mathcal{S}$ is strictly reducible. We conjecture that the converse is also true.