Stick number of 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 $s(K)\le 2c(K)$. Later, Huh and Oh utilized the arc index $\alpha (K)$ to present a more precise upper bound $s(K)\le \frac{3}{2}c(K)+\frac{3}{2}$. Furthermore, Kim, No and Oh found an upper bound on the equilateral stick number $s_{=}(K)$ as follows; $s_{=}(K)\le 2c(K)+2$. 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; $$s(G)\le \frac{3}{2}c(G)+2e+\frac{3b}{2}-\frac{v}{2},$$ $$s_{=}(G)\le 2c(G)+2e+2b-k,$$ 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.