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arXiv · 2508.02108

Monotone Paths on Acyclic 3-Regular Graphs

François Clément·Dan Guyer
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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 $\varphi^n$.

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BibTeXRIS

François Clément, Dan Guyer. 2025-08-04. Monotone Paths on Acyclic 3-Regular Graphs. https://arxiv.org/abs/2508.02108

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