In this work we show that with high probability the chromatic number of a graph sampled from the random regular graph model Gn,d for d = o(n 1/5) is concentrated in two consecutive values, thus extending a previous result of Achlioptas and Moore. This concentration phenomena is very similar to that of the binomial random graph model G(n, p) with p = d. Our proof is largely based on ideas of Alon n and Krivelevich who proved this two-point concentration result for G(n, p) for p = n −δ where δ> 1/2. The main tool used to derive such a result is a careful analysis of the distribution of edges in Gn,d, relying both on the switching technique and on bounding the probability of exponentially small events in the configuration model. 1

For an increasing monotone graph property P the local resilience of a graph G with respect to P is the minimal r for which there exists of 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 possesses P. 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 G(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, Gn,d, with respect to being Hamiltonian is at least (1−ε)d/6. We also prove that for the Binomial random graph model G(n, p), for every positive ε> 0 and large enough values of K, if p> K ln n n then with high probability the local resilience of G(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. 1

We consider the spectral and algorithmic aspects of the problem of finding a Hamiltonian cycle in a graph. We show that a sufficient condition for a graph being Hamiltonian is that the eigenvalues of the combinatorial Laplacian are sufficiently close to the average degree of the graph. An algorithm is given for the problem of finding a Hamiltonian cycle in graphs with bounded spectral gaps which has complexity of order n c ln n. 1

Recall that in the last lecture we looked at the problem of isoperimetric inequalities in the hypercube, Qn. Our notion of boundary was that of vertex boundary, defined by δ(G) = {v ̸ ∈ S | v ∼ u, u ∈ S}. We found that when trying to minimize the vertex boundary while holding the size of our set S fixed, the (near) Hamming balls characterize those S with minimal vertex boundary. In this lecture, we consider a different notion of boundary based on edges. We define the edge boundary of S to be ∂(S) = {{u, v} | u ∈ S, v ̸ ∈ S}. This is exactly the set of edges required to disconnect S from any vertex not in S. 1.1 Four Problems We shall look at four related isoperimetric problems which all utilize the concept of edge boundary. 1.1.1 Question 1 We begin with the simplest question: Given a fixed positive integer m, what is the smallest edge boundary for a set of m vertices? We may formalize this question by defining g(m) = min

We say that a graph G = (V, E) on n vertices is a β-expander for some constant β> 0 if every U ⊆ V of cardinality |U | ≤ n satisfies |NG(U) | ≥ β|U | where NG(U) denotes the neighborhood of U. We 2 explore the process of uniformly at random deleting vertices of a β-expander with probability n −α for some constant α> 0. Our main result implies that as n tends to infinity, the deletion process performed on a β-expander graph of bounded degree will result with high probability in a graph composed of a giant component containing n − o(n) vertices which is itself an expander graph, and small constant size components. We proceed by applying the main result to expander graphs with a positive spectral gap. In the particular case of (n, d, λ)-graphs, which are such expanders, we compute the values of α, under additional constraints on the graph, for which with high probability the resulting graph will stay connected, or will be composed of a giant component and isolated vertices. As a graph sampled from the uniform probability space of d-regular graphs with hight probability meets all of these constraints, this result strengthens a recent result due to Greenhill, Holt, and Wormald [6] who prove a similar theorem for Gn,d. We conclude by showing that performing the deletion process with the prescribed deletion probability on expander graphs that expand sub-linear sets by an unbounded expansion ratio, with high probability results in an expander graph. 1