Kelmans–Seymour conjecture

One way of restating this result is that, in these graphs, it is always possible to perform a sequence of edge contraction operations so that the resulting graph contains a K5 subdivision.

The Kelmans–Seymour conjecture states that, with a higher order of connectivity, these contractions are not required.

An earlier conjecture of Gabriel Andrew Dirac (1964), proven in 2001 by Wolfgang Mader, states that every n-vertex graph with at least 3n − 5 edges contains a subdivision of K5.

[3][4][5] In 2016, a proof of the Kelmans–Seymour conjecture was claimed by Xingxing Yu of the Georgia Institute of Technology and his Ph.D. students Dawei He and Yan Wang.

[1] A sequence four papers proving this conjecture appeared in Journal of Combinatorial Theory, Series B.