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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 2 edges(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)
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)
In this article we study Hamilton cycles in sparse pseudo-random graphs. We prove that if the second largest absolute value of an eigenvalue of a d-regular graph G on n vertices satisfies —————————————————— ðlog log nÞ 2 1000 log n ðlog log log nÞ d and n is large enough, then G is Hamiltonian. We also show how our main result can be used to prove that for(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 r-degenerate 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)
Let P be an optimization problem, and let A be an approximation algorithm for P. The domination ratio domr(A, n) is the maximum real q such that the solution x(I) obtained by A for any instance I of P of size n is not worse than at least a fraction q of the feasible solutions of I. We describe a deterministic, polynomial time algorithm with domination ratio(More)
Let G be a graph on n vertices and suppose that at least n 2 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)