Topological Ramsey Theorem for Complete Bipartite Graphs
Copyright policy )Topological Ramsey Theorem for Complete Bipartite Graphs
An embedding of a graph into a space is linear if each edge is a straight line segment. In 1991, \textit{S. Negami} [Trans. Am. Math. Soc. 324, No. 2, 527-541 (1991; Zbl 0721.57004)] showed that for any given knot, link, or spatial graph there is a sufficiently large complete graph \(K_ n\) such that every linear embedding of \(K_ n\) into a space always contains that knot, link, or spatial graph. This paper generalizes this result to cover complete bipartite graphs. The results for complete multipartite graphs and for complete graphs are obtained as corollaries.
complete graphs, link, Generalized Ramsey theory, Ramsey theorem, spatial graph, complete multipartite graphs, Planar graphs; geometric and topological aspects of graph theory, Theoretical Computer Science, complete bipartite graphs, embedding, Computational Theory and Mathematics, knot, Knots and links in the \(3\)-sphere, Discrete Mathematics and Combinatorics, Relations of low-dimensional topology with graph theory
complete graphs, link, Generalized Ramsey theory, Ramsey theorem, spatial graph, complete multipartite graphs, Planar graphs; geometric and topological aspects of graph theory, Theoretical Computer Science, complete bipartite graphs, embedding, Computational Theory and Mathematics, knot, Knots and links in the \(3\)-sphere, Discrete Mathematics and Combinatorics, Relations of low-dimensional topology with graph theory
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