We consider the following probabilistic model of a graph on n labeled vertices. First choose a random graph G(n, 1/2) and then choose randomly a subset Q of vertices of size k and force it to be aâ€¦ (More)

For a graph H and an integer n, the TurÃ¡n 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 itsâ€¦ (More)

A graphG = (V ,E) on n vertices is( , )-regular if its minimal degree is at least n, and for every pair of disjoint subsets S, T âŠ‚ V of cardinalities at least n, the number of edges e(S, T )â€¦ (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 growsâ€¦ (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 theâ€¦ (More)

Random graphs have proven to be one of the most important and fruitful concepts in modern Combinatorics and Theoretical Computer Science. Besides being a fascinating study subject for their own sake,â€¦ (More)

In this article we study Hamilton cycles in sparse pseudorandom graphs. We prove that if the second largest absolute value of an eigenvalue of a d-regular graph G on n vertices satisfies