Splittable Spanning Trees and Balanced Forests in Dense Random Graphs

We consider the probability that a spanning tree chosen uniformly at random from a graph can be partitioned into a fixed number $k$ of trees of equal size by removing $k-1$ edges. In that case, the spanning tree is called {\em splittable}. Splittable spanning trees are useful in algorithms for sampling {\em balanced forests}, forests whose components are of equal size, and for sampling partitions of a graph into components of equal size, with applications in redistricting, network algorithms, and image decomposition. Cannon et al.~recently showed that spanning trees on grid and grid-like graphs on $n$ vertices are splittable into $k$ equal sized components with probability at least $n^{-2k}$, leading to the first rigorous sampling algorithm for balanced forests in any class of graphs. Focusing on the complementary case of dense random graphs, we show that random spanning trees have inverse polynomial probability of being splittable; specifically, a random spanning tree is splittable with probability at least $n^{(-k/2)}$ for both the $G(n,p)$ and $G(n,m)$ models when $p = Ω(1/\log n)$, giving the first dense class of graphs where partitions of equal size can be sampled efficiently. In addition, we present an infinite family of graphs with properties that have been conjectured to ensure splittability (i.e., Hamiltonian subgraphs of the triangular lattice) and prove that random spanning trees are not splittable with more than exponentially small probability. As a consequence, we show that a family of widely-used Markov chain algorithms for sampling equal-size partitions will fail on this family of graphs if their state spaces are restricted to equal-size partitions. Moreover, we show these algorithms will be inefficient if their state spaces are generalized to include any unbalanced partitions, suggesting barriers for sampling balanced partitions in sparse graphs.