Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, December 12, 2017, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Jerri Nummenpalo

Sparse Kneser graphs are Hamiltonian

For integers k≥1 and n≥2k+1, the Kneser graph K(n,k) is the graph whose vertices are the k-element subsets of {1,…,n} and whose edges connect pairs of subsets that are disjoint. The Kneser graphs of the form K(2k+1,k) are also known as the odd graphs. We settle an old problem due to Meredith, Lloyd, and Biggs from the 1970s, proving that for every k≥3, the odd graph K(2k+1,k) has a Hamilton cycle. This and a known conditional result due to Johnson imply that all Kneser graphs of the form K(2k+2^a,k) with k≥3 and a≥0 have a Hamilton cycle. We also prove that K(2k+1,k) has doubly exponentially many different Hamilton cycles for k≥6. Our proofs are based on a reduction of the Hamiltonicity problem in the odd graph to the problem of finding a spanning tree in a suitably defined hypergraph on Dyck words. This is joint work with Torsten Mütze and Bartosz Walczak.

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