Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, December 08, 2016, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Miloš Trujić

Local resilience of an almost spanning k-cycle in sparse random graphs

Pósa and Seymour conjectured that for any k > 1, a graph with minimum degree at least kn/(k + 1) contains the k-th power of a Hamilton cycle, namely, a Hamilton cycle where additionally between every pair of vertices at distance at most k, there is an edge. Only much later, after appearance of tools such as Szemerédi's Regularity Lemma and the Blow-up Lemma, Komlós, Sárközy, and Szemerédi confirmed the Pósa-Seymour conjecture for large graphs.

We extend the result of Komlós et. al. to random graphs by showing that for all k > 1 and alpha, epsilon > 0, and p > (log n/n)^(1/k), any subgraph of a random graph G(n, p) with minimum degree (k/(k + 1) + alpha)np, w.h.p. contains the k-th power of a cycle on (1 - epsilon)n vertices, thus improving a recent result of Noever and Steger for k = 2, and generalising it to any k. Our result is almost optimal for the following reasons. The constant k/(k+1) in the bound on the minimal degree cannot be improved. When p < n^(-1/k), G(n, p) will not contain the k-th power of a long cycle w.h.p. Finally, just by deleting an epsilon fraction of the edges touching each vertex, it is easy for an adversary to make sure that some vertices are not contained in the k-th power of a cycle, which shows that one cannot hope for an improvement of the result where k-th power of a Hamilton cycle is obtained instead of a cycle on (1 - epsilon)n vertices.

This is joint work with Angelika Steger and Nemanja Škorić.

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