Conway's thrackle conjecture

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A thrackle is an embedding (in the plane) of a graph, such that each edge is a Jordan arc and every pair of edges meet once. Edges may either meet at a common endpoint, or, if they have no endpoints in common, at a point in their interiors. In the latter case, the crossing must be transverse.^[1]

John H. Conway has conjectured that in any thrackle the number of edges is at most equal to the number of vertices. Conway himself uses the terminology paths and spots (for edges and vertices respectively), so Conway's thrackle conjecture was originally stated in the form every thrackle has at least as many spots as paths.

It has been proved that every cycle graph other than C₄ has a thrackle embedding, which shows that the conjecture is sharp. That is, there are thrackles having the same number of spots as paths. At the other extreme, the worst case scenario is that the number of spots is twice the number of paths; which is also attainable.

Lovász et al. (1997) proved that every bipartite thrackle is a planar graph, although not drawn in a planar way. As a consequence, every such graph with n vertices has at most 2n − 4 edges. Based on the same ideas, they show that every thrackle has at most 2n − 4 edges.

[edit] References

1. ^ L. Lovász, J. Pach, and M. Szegedy, On Conway's thrackle conjecture, Discrete Comput. Geom. 18 (1997), 369--376.