Towards Crossing-Free Hamiltonian Cycles in Simple Drawings of Complete Graphs

03/27/2023

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by   Oswin Aichholzer, et al.

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It is a longstanding conjecture that every simple drawing of a complete graph on n≥ 3 vertices contains a crossing-free Hamiltonian cycle. We confirm this conjecture for cylindrical drawings, strongly c-monotone drawings, as well as x-bounded drawings. Moreover, we introduce the stronger question of whether a crossing-free Hamiltonian path between each pair of vertices always exists.