Monotone Paths on Acyclic 3-Regular Graphs

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 $(1+\phi {)}^n$, with $\phi$ 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 ${\phi }^n$.