Monotone Paths on Acyclic 3-Regular Graphs

TL;DR

This paper improves bounds on the number of monotone paths in acyclic 3-regular graphs, advancing understanding related to the behavior of the simplex method on 3-polytopes.

Contribution

It provides a tighter upper bound of c·1.6779^n for a broader class of graphs, surpassing previous bounds involving the golden ratio.

Findings

Number of monotone paths bounded by c·1.6779^n

Improved bounds extend to larger graph families

Conjecture that extremal case is approximately φ^n

Abstract

Motivated by trying to understand the behavior of the simplex method, Athanasiadis, De Loera and Zhang provided upper and lower bounds on the number of the monotone paths on 3-polytopes. For simple 3-polytopes with $2n$ vertices, they showed that the number of monotone paths is bounded above by $(1+\varphi)^n$, with $\varphi$ being the golden ratio. We improve the result and show that for a larger family of graphs the number is bounded above by $c \cdot 1.6779^n$ for some universal constant $c$. Meanwhile, the best known construction and conjectured extremizer is approximately $\varphi^n$.