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In highly connected networks, there's always a loop

Mathematicians show that graphs of a certain common type must contain a route that visits each point exactly once.

In 1976, the Hungarian mathematician Lajos Pósa proved that certain graphs built by randomly drawing edges were virtually guaranteed to contain Hamiltonian cycles. That changed in March 2023, when Sudakov, his student David Munhá Correia, and Stefan Glock of the University of Passau improved on the 2002 result, showing that a slightly larger class of expander graphs must have Hamiltonian cycles. Montgomery had tried to prove Sudakov and Krivelevich’s conjecture as part of his doctorate at Cambridge in the early 2010s, but had given up because he thought he didn’t have the right tools to tackle the problem.

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