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

Date and Time: Tuesday, May 09, 2017, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Miloš Trujić

Local resilience of sparse random graph properties at the threshold

Two of the most famous examples of studying containment of spanning subgraphs in sparse random graphs are the results of Erdős and Rényi (1966) stating that a random graph w.h.p. has a perfect matching as soon as p > (log n + omega(1))/n, and that of Komlós and Szemerédi (1983), Bollobás (1984), and Ajtai, Komlós, and Szemerédi (1985) stating that a random graph is w.h.p. Hamiltonian as soon as p > (log n + loglog n + omega(1))/n.

Sudakov and Vu (2008) introduced the concept of local resilience in random graphs, namely the questions of the type "How many edges touching each vertex can one remove and still claim that the remaining graph satisfies the desired property?". Already in this paper they show that a random graph for p > Clog n/n w.h.p. has a perfect matching even after the removal of at most (1/2 - o(1))np edges touching each vertex. A couple of years later Lee and Sudakov (2012) showed that a random graph for p > Clog n/n is w.h.p. Hamiltonian even after the removal of at most (1/2 - o(1))np edges touching each vertex.

In this result we show that for any beta > 0, and p > (log n + omega(1))/n, a random graph w.h.p. has a perfect matching even after the removal of at most (1/2 - beta)-fraction of the edges touching each vertex. Moreover, for p > (log n + loglog n + omega(1))/n, a random graph is w.h.p. Hamiltonian even after the removal of at most (1/2 - beta)-fraction of the edges touching each vertex.

This is joint work with Rajko Nenadov and Angelika Steger.

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