Comparing the numbers of subforests and subgraph-degree-tuples
Sergei Shteiner, Pavel Shteyner
TL;DR
The paper connects the combinatorics of row-column-sums for square tridiagonal $0$-$1$ matrices with the acyclic-subgraph counts of complete ladder graphs, revealing the equality with the OEIS sequence $A022026$. It generalizes this correspondence to $k$-colored grids and broader sparsity patterns, and derives a graph-theoretic framework (FED/FLD) linking subforests to degree-tuples via biadjacency representations. It proves key equalities for bipartite graphs, notably complete bipartite, complete tridiagonal, cactus, and generalized book graphs, while formulating strong conjectures: bipartite graphs are precisely those with $|\mathcal{F}(G)|=|\mathcal{D}(G)|$, and non-bipartite graphs exhibit strict inequality. The work develops factorization techniques over bridges and articulation points, enabling reductions and explicit formulas in several graph families, and lays a foundation for extending the correspondence to broader matrix entry sets and sparsity patterns with potential algorithmic implications.
Abstract
We enumerate the row-column-sums of all square tridiagonal $(0,1)$-matrices and prove that their count coincides with OEIS A022026 $-$ the number of acyclic subgraphs of the complete $2\times n$ grid graph. We then extend this correspondence in two independent directions: 1. admitting larger sets of matrix entries, and 2. relaxing the tridiagonal support to broader prescribed sparsity patterns. The latter leads us to conjecture that, for any bipartite graph $G$, the number of its acyclic subgraphs equals the number of degree sequences realized by subgraphs of $G$. Moreover, for any non-bipartite graph, the former should be strictly smaller than the latter. We discuss several general approaches and prove these hypotheses for cactus graphs and generalized book graphs.