Generalization of the Conway--Gordon theorem and intrinsic linking on complete graphs
Generalization of the Conway--Gordon theorem and intrinsic linking on complete graphs
复制标题
康威的推广--戈登定理和完全图上的内在联系
DOI：
10.1007/s00026-021-00536-5
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发表时间：
2021
影响因子：
0.5
通讯作者：
Ryo Nikkuni
中科院分区：
文献类型：
--
作者：
Hiroko Morishita;Ryo Nikkuni
Conway和Gordon证明了对于每个顶点数为6的空间完全图，其所有组成二分量链的连接数之和为奇数; Kazakov和Korablev证明了对于每个顶点数大于6的空间完全图，其所有组成二分量哈密尔顿链的连接数之和为偶数。本文证明了对于顶点数大于6的空间完全图，所有二分量Hamilton链上的链数平方之和可以用所有三角-三角组成链上的和来确定.作为应用，我们证明了如果顶点数足够大，则每个空间完全图都包含一个两分量Hamilton链，其链数的绝对值是任意大的.最后给出了直线空间完全图的一些应用。
Conway and Gordon proved that for every spatial complete graph on six vertices, the sum of the linking numbers over all of the constituent two-component links is odd, and Kazakov and Korablev proved that for every spatial complete graph with arbitrary number of vertices greater than six, the sum of the linking numbers over all of the constituent two-component Hamiltonian links is even. In this paper, we show that for every spatial complete graph whose number of vertices is greater than six, the sum of the square of the linking numbers over all of the two-component Hamiltonian links is determined explicitly in terms of the sum over all of the triangle–triangle constituent links. As an application, we show that if the number of vertices is sufficiently large then every spatial complete graph contains a two-component Hamiltonian link whose absolute value of the linking number is arbitrary large. Some applications to rectilinear spatial complete graphs are also given.