A Contribution to Guy’s Conjecture

Abstract

Let K_n denote the complete graph consisting of n vertices, every pair of which forms an edge. We want to know the least possible number of the intersections, when we draw the graph K_n on the plane or on the sphere using continuous arcs for edges. In a paper published in 1960, Richard K. Guy conjectured that the least possible number of the intersections is ${1 \over {64}}{\left( {n - 1} \right)^2}{\left( {n - 3} \right)^2}$ if n is odd, or ${1 \over {64}}n{\left( {n - 2} \right)^2}\left( {n - 4} \right)$ if n is even. A virgin road V_n is a drawing of a Hamiltonian cycle in K_n consisting of n vertices and n edges such that no other edge-representing arcs cross V_n. A drawing of K_n is called virginal if it contains a virgin road. All drawings considered in this paper will be virginal with respect to a fixed virgin road V_n. We will present a certain drawing of a subgraph of K_n, for each n(≥ 5), which is “characteristic” in the sense that any minimal virginal drawing of K_n containing this subdrawing satisfies Guy’s conjecture.