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)
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 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)
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)
We study two problems related to the existence of Hamilton cycles in random graphs. The first question relates to the number of edge disjoint Hamilton cycles that the random graph G n,p contains. δ(G)/2 is an upper bound and we show that if p ≤ (1 + o(1)) ln n/n then this upper bound is tight whp. The second question relates to how many edges can be… (More)
It is proven that for k ≥ 4 and n > k every k-color-critical graph on n vertices has at least k−1 2 + k−3 2(k 2 −2k−1) n edges, thus improving a result of Gallai from 1963.
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 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)