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On Conway’s thrackle conjecture. (English) Zbl 0892.05017
A thrackle is a graph drawn in the plane with its edges represented by Jordan arcs so that any two distinct arcs either meet at exactly one common vertex or cross at exactly one point interior to both arcs. J. H. Conway’s thrackle conjecture is that the number of edges of a thrackle is at most the number of its vertices. The present authors show that every thrackleable bipartite graph is planar, and use this result to show that a thrackle having \(n\) vertices has at most \(2n-3\) edges. They also consider some related problems and generalizations.

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
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