Stick number of spatial graphs

TL;DR

This paper extends bounds on stick numbers from knots to spatial graphs, providing new upper bounds based on crossing number, edges, vertices, and component counts, advancing understanding of spatial graph complexity.

Contribution

It introduces definitions of stick number and equilateral stick number for spatial graphs and establishes their upper bounds, generalizing previous knot results.

Findings

Derived upper bounds for stick number of spatial graphs

Established bounds depend on crossing number, edges, and vertices

Generalized knot bounds to spatial graphs

Abstract

For a nontrivial knot $K$, Negami found an upper bound on the stick number $s(K)$ in terms of its crossing number $c(K)$ which is $s(K) \leq 2 c(K)$. Later, Huh and Oh utilized the arc index $\alpha(K)$ to present a more precise upper bound $s(K) \leq \frac{3}{2} c(K) + \frac{3}{2}$. Furthermore, Kim, No and Oh found an upper bound on the equilateral stick number $s_{=}(K)$ as follows; $s_{=}(K) \leq 2 c(K) +2$. As a sequel to this research program, we similarly define the stick number $s(G)$ and the equilateral stick number $s_{=}(G)$ of a spatial graph $G$, and present their upper bounds as follows; $$ s(G) \leq \frac{3}{2} c(G) + 2e + \frac{3b}{2} -\frac{v}{2}, $$ $$ s_{=}(G) \leq 2 c(G) + 2e + 2b - k, $$ where $e$ and $v$ are the number of edges and vertices of $G$, respectively, $b$ is the number of bouquet cut-components, and $k$ is the number of non-splittable components.