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) <= 2c(K). Later, Huh and Oh utilized the arc index a(K) to present a more precise upper bound s(K) <= 3/2 c(K)+3/2. Furthermore, Kim, No and Oh found an upper bound on the equilateral stick number s=(K) as follows; s=(K) <= 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) <= 3/2 c(G) + 2e + 3b/2-v/2, s=(G) <= 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.