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Generalization of the Conway-Gordon theorem and intrinsic linking on complete graphs. (English) Zbl 1475.57030

This paper proves a theorem about two-component links that contain all vertices of an embedding of complete graphs: It is shown 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 (i.e. links containing all vertices) is determined explicitly in terms of the sum over all of the triangle-triangle constituent links. Corollaries for graphs with large number of vertices are proven and consequences for rectilinear (i.e. piecewise linear) embeddings of complete graphs are presented.

MSC:

57M15 Relations of low-dimensional topology with graph theory
57K10 Knot theory
05C10 Planar graphs; geometric and topological aspects of graph theory
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