K-connectivity in random undirected graphs

Abstract

This paper concerns vertex connectivity in random graphs. We present results bounding the cardinality of the biggest k-block in random graphs of the G n, p model, for any constant value of k. Our results extend the work of Erdös and Rényi and Karp and Tarjan. We prove here that G n, p , with p⩾ c n , has a giant k-block almost surely, for any constant k>0. The distribution of the size of the giant k-block is examined. We provide bounds on this distribution which are very nearly tight. We furthermore prove here that the cardinality of the biggest k-block is greater than n-log n, with probability at least 1−1/( n 2log n), for p⩾ c n and c> k+3.