Robustness of Pósa’s conjecture Master Thesis

The kth power of a cycle is obtained by adding an edge between all pairs of vertices whose distance on the cycle is at most k. In 1962, Pósa conjectured that a graph G on n vertices contains a square of a Hamilton cycle if it has minimum degree δ(G) ≥ 3 n, and, 11 years later, Seymour claimed that δ(G) ≥ k k+1 n is sufficient for the appearance of a kth power of a Hamilton cycle for any k ≥ 2. The so-called Pósa-Seymour conjecture could be resolved for sufficiently large graphs by Komlós, Sárközy, and Szemerédi in 1998 and the complete (k + 1)-partite graph having k sets of size n−1 k+1 witnesses optimality of the constant k k+1 . The main purpose of the thesis is to extend this statement to the random graph setting. We prove that for every α, β, ε ∈ (0, 1) and p2 ≥ n−1+ε a.a.s. every subgraph of G(n, p) with minimum degree δ(G) ≥ ( 2 3 + α ) np and minimum codegree δ2(G) ≥ βnp2 contains a square of a Hamilton cycle. A simple application of the First Moment method, which yields that G(n, p) a.a.s. does not contain any square of a Hamilton cycle for p2 n−1, reveals almost optimality of the edge probability in this result and the tightness of Pósa’s conjecture shows that the constant in the minimum degree is asymptotically best possible. Moreover, the assumption on the codegrees cannot be omitted, as without this condition every square of a Hamilton cycle can easily be obstructed by deleting all edges inside the neighborhood of one vertex.