Sonny Ben-Shimon

On the Resilience of Hamiltonicity and Optimal Packing of Hamilton Cycles in Random Graphs

Let k=(k1,…,kn) be a sequence of n integers. For an increasing monotone graph property 𝒫 we say that a base graph G=([n],E) is k-resilient with respect to 𝒫 if for every subgraph H⊆G such that dH(i)≤ki for every 1≤i≤n the graph G-H possesses 𝒫. This notion naturally extends the idea of the local resilience of graphs recently initiated by Sudakov and Vu. In this paper we study the k-resilience of a typical graph from 𝒢(n,p) with respect to the Hamiltonicity property, where we let p range over all values for which the base graph is expected to be Hamiltonian. Considering this generalized approach to the notion of resilience our main result implies several corollaries which improve on the best known bounds of Hamiltonicity related questions. For one, it implies that for every positive e>0 and large enough values of K, if p>Klnnn, then with high probability the local resilience of 𝒢(n,p) with respect to being Hamiltonian is at least (1-e)np/3, improving on the previous bound for this range of p. Another impli...

Local resilience and hamiltonicity maker–breaker games in random regular graphs

For an increasing monotone graph property the local resilience of a graph G with respect to is the minimal r for which there exists a subgraph H ⊆ G with all degrees at most r, such that the removal of the edges of H from G creates a graph that does not possess . This notion, which was implicitly studied for some ad hoc properties, was recently treated in a more systematic way in a paper by Sudakov and Vu. Most research conducted with respect to this distance notion focused on the binomial random graph model (n, p) and some families of pseudo-random graphs with respect to several graph properties, such as containing a perfect matching and being Hamiltonian, to name a few. In this paper we continue to explore the local resilience notion, but turn our attention to random and pseudo-random regular graphs of constant degree. We investigate the local resilience of the typical random d-regular graph with respect to edge and vertex connectivity, containing a perfect matching, and being Hamiltonian. In particular, we prove that for every positive ϵ and large enough values of d, with high probability, the local resilience of the random d-regular graph, n, d, with respect to being Hamiltonian, is at least (1−ϵ)d/6. We also prove that for the binomial random graph model (n, p), for every positive ϵ > 0 and large enough values of K, if p >

Hitting time results for Maker-Breaker games

\frac{K\ln n}{n}

A note on regular Ramsey graphs

then, with high probability, the local resilience of (n, p) with respect to being Hamiltonian is at least (1−ϵ)np/6. Finally, we apply similar techniques to positional games, and prove that if d is large enough then, with high probability, a typical random d-regular graph G is such that, in the unbiased Maker–Breaker game played on the edges of G, Maker has a winning strategy to create a Hamilton cycle.

Random regular graphs of non-constant degree: Concentration of the chromatic number

We study Maker-Breaker games played on the edge set of a random graph. Specifically, we analyze the moment a typical random graph process first becomes a Makers win in a game in which Makers goal is to build a graph which admits some monotone increasing property \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \begin{align*}\mathcal{P}\end{align*} \end{document} **image** . We focus on three natural target properties for Makers graph, namely being k -vertex-connected, admitting a perfect matching, and being Hamiltonian. We prove the following optimal hitting time results: with high probability Maker wins the k -vertex connectivity game exactly at the time the random graph process first reaches minimum degree 2k; with high probability Maker wins the perfect matching game exactly at the time the random graph process first reaches minimum degree 2; with high probability Maker wins the Hamiltonicity game exactly at the time the random graph process first reaches minimum degree 4. The latter two statements settle conjectures of Stojakovic and Szabo. We also prove generalizations of the latter two results; these generalizations partially strengthen some known results in the theory of random graphs.