2018/06/25 by Lee, Minjung, No, Sungjong, Oh, Seungsang
#57M25 #57M27 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.1806.09716
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) ≤ 2 c(K). Later, Huh and Oh utilized the arc index α(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) ≤ 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) ≤ (3)/(2) c(G) + 2e + (3b)/(2) -(v)/(2), s=(G) ≤ 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.