Given a sequence of nonnegative real numbers A,, A,, . . . which sum to 1, we consider random graphs having approximately Ain vertices of degree i. Essentially, we show that if C i(i -2)A, > 0, then such graphs almost surely have a giant component, while if C i(i -2)A, < 0, then almost surely all components in such graphs are small. We can apply these results to G n , p , Gn,M, and other well-known models of random graphs. There are also applications related to the chromatic number of sparse random graphs.

Given a sequence of non-negative r e a l n umbers 0 1 : : :which s u m t o 1 , w e consider a random graph having approximately i n vertices of degree i. In 12] the authors essentially show that if P i(i ; 2) i > 0 then the graph a.s. has a giant component, while if P i(i ; 2) i < 0 then a.s. all components in the graph are small. In this paper we analyze the size of the giant component in the former case, and the structure of the graph formed by deleting that component. We determine 0 0 0 1 : : : such that a.s. the giant component, C, h a s n + o ( n) v ertices, and the structure of the graph remaining after deleting C is basically that of a random graph with n 0 = n ; j Cj vertices, and with 0 i n 0 of them of degree i.

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ABSTRACT:We describe a technique for determining the thresholds for the appearance of cores in random structures. We use it to determine (i) the threshold for the appearance of a k-core in a random r-uniform hypergraph for all r, k ≥ 2, r + k > 4, and (ii) the threshold for the pure literal rule to find a satisfying assignment for a random instance of r-SAT, r ≥ 3.

ABSTRACT:We consider random systems of linear equations over GF(2) in which every equation binds k variables. We obtain a precise description of the clustering of solutions in such systems. In particular, we prove that with probability that tends to 1 as the number of variables, n, grows: for every pair of solutions σ , τ , either there exists a sequence of solutions starting at σ and ending at τ such that successive solutions have Hamming distance O(log n), or every sequence of solutions starting at σ and ending at τ contains a pair of successive solutions with distance (n). Furthermore, we determine precisely which pairs of solutions are in each category. Key to our results is establishing the following high probability property of cores of random hypergraphs which is of independent interest. Every vertex not in the r-core of a random k-uniform hypergraph can be removed by a sequence of O(log n) steps, where each step amounts to removing one vertex of degree strictly less than r at the time of removal.

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