Michael Krivelevich

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We consider the following probabilistic model of a graph on n labeled vertices. Ž . First choose a random graph G n, 1r2 , and then choose randomly a subset Q of vertices of size k and force it to be a clique by joining every pair of vertices of Q by an edge. The problem is to give a polynomial time algorithm for finding this hidden clique almost surely for(More)
Let P be a property of graphs. An -test for P is a randomized algorithm which, given the ability to make queries whether a desired pair of vertices of an input graph G with n vertices are adjacent or not, distinguishes, with high probability, between the case of G satisfying P and the case that it has to be modified by adding and removing more than n edges(More)
A code is locally testable if there is a way to indicate with high probability that a vector is far enough from any codeword by accessing only a very small number of the vector's bits. We show that the Reed-Muller codes of constant order are locally testable. Specifically, we describe an efficient randomized algorithm to test if a given vector of length(More)
Let G be a graph on n vertices and suppose that at least n edges have to be deleted from it to make it k-colorable. It is shown that in this case most induced subgraphs of G on ck ln k/ 2 vertices are not k-colorable, where c > 0 is an absolute constant. If G is as above for k = 2, then most induced subgraphs on (ln(1/ )) b are non-bipartite, for some(More)
New upper bounds on the rate of low-density parity-check (LDPC) codes as a function of the minimum distance of the code are derived. The bounds apply to regular LDPC codes, and sometimes also to right-regular LDPC codes. Their derivation is based on combinatorial arguments and linear programming. The new bounds improve upon the previous bounds due to(More)
An edge-colored graph G is rainbow edge-connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow edgeconnected. We prove that if G has n vertices and minimum degree δ then rc(G) < 20n/δ.(More)
For a graph H and an integer n, the Turán number ex(n,H) is the maximum possible number of edges in a simple graph on n vertices that contains no copy of H . H is rdegenerate if every one of its subgraphs contains a vertex of degree at most r. We prove that, for any fixed bipartite graph H in which all degrees in one colour class are at most r, ex(n,H)(More)
For every fixed integers r, s satisfying 2 ≤ r < s there exists some = (r, s) > 0 for which we construct explicitly an infinite family of graphs Hr,s,n, where Hr,s,n has n vertices, contains no clique on s vertices and every subset of at least n1− of its vertices contains a clique of size r. The constructions are based on spectral and geometric techniques,(More)
Random d-regular graphs have been well studied when d is fixed and the number of vertices goes to infinity. We obtain results on many of the properties of a random d-regular graph when d = d n grows more quickly than √n. These properties include connectivity, hamiltonicity, independent set size, chromatic number, choice number, and the size of the second(More)