Increasing spanning forests in graphs and simplicial complexes

TL;DR

This paper studies increasing spanning forests in graphs and simplicial complexes, providing new combinatorial proofs, bounds on coefficients, and generalizations including cage-free complexes and pattern-avoiding label sequences.

Contribution

It offers new combinatorial proofs of known results, extends the theory to simplicial complexes and multigraphs, and introduces the concept of cage-free complexes.

Findings

The generating function for increasing spanning forests has all nonpositive integral roots.

The polynomial equals the chromatic polynomial under perfect elimination order.

Bounds on coefficients are established using broken circuits.

Abstract

Let G be a graph with vertex set {1,...,n}. A spanning forest F of G is increasing if the sequence of labels on any path starting at the minimum vertex of a tree of F form an increasing sequence. Hallam and Sagan showed that the generating function ISF(G,t) for increasing spanning forests of G has all nonpositive integral roots. Furthermore they proved that, up to a change of sign, this polynomial equals the chromatic polynomial of G precisely when 1,...,n is a perfect elimination order for G. We give new, purely combinatorial proofs of these results which permit us to generalize them in several ways. For example, we are able to bound the coefficients of ISF(G,t) using broken circuits. We are also able to extend these results to simplicial complexes using the new notion of a cage-free complex. A generalization to labeled multigraphs is also given. We end by exploring spanning forests where the increasing condition is replaced by having the label sequences avoid the patterns 231, 312, and 321.