A Contribution to Guy's Conjecture  

作　　者：Yukio MATSUMOTO Yoshikazu MATSUTANI Angel MONTESINOS-AMILIBIA Masami ODA Shuichi OHKI Tsuyoshi SAKAI 

机构地区：[1]Department of Mathematics,Gakushuin University,Mejiro,Toshima-ku,Tokyo 171-8588,Japan [2]3-21-1-901 Minamimachi,Kokubunji,Tokyo 185-0021,Japan [3]Calle Rio Alberche,8.05260 Cebreros,Spain [4]Tsuda University,2-1-1,Tsuda-cho,Kodaira,Tokyo 187-8577,Japan [5]Tokyo Japanese Language Educational Center,3-22-7,Kita Shinjuku,Shinjuku-ku,Tokyo 169-0074,Japan [6]Department of Mathematics,College of Humanities and Sciences,Nihon University,Setagaya-ku,Tokyo 156-0045,Japan

出　　处：《Acta Mathematica Sinica,English Series》2022年第10期1856-1886,共31页数学学报（英文版）

基　　金：supported by JSPS Grant KAKENHI 17H01091。

摘　　要：Let Kdenote 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 Kon 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 1/64(n-1)^(2)(n-3)^(2) if n is odd,or 1/64 n(n-2)^(2)(n-4)if n is even.A virgin road V_(n)is a drawing of a Hamiltonian cycle in Kconsisting of n vertices and n edges such that no other edge-representing arcs cross V.A drawing of Kis 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.We will present a certain drawing of a subgraph of K,for each n(≥5),which is"characteristic"in the sense that any minimal virginal drawing of Kcontaining this subdrawing satisfies Guy’s conjecture.

关 键 词：Complete graph virginal drawing Guy’s conjecture 

分 类 号：O157.5[理学—数学]