2008/04/08 by Noga Alon, Alon, Noga, Asaf Nussboim +1
Computer Science · Mathematics · #05C80 #Combinatorics (math.CO) #Complexity and Algorithms in Graphs #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.0804.1268
openalex publication_date 2008/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the k-wise independent relaxation of the usual model G(N,p) of random graphs where, as in this model, N labeled vertices are fixed and each edge is drawn with probability p, however, it is only required that the distribution of any subset of k edges is independent. This relaxation can be relevant in modeling phenomena where only k-wise independence is assumed to hold, and is also useful when the relevant graphs are so huge that handling G(N,p) graphs becomes infeasible, and cheaper random-looking distributions (such as k-wise independent ones) must be used instead. Unfortunately, many well-known properties of random graphs in G(N,p) are global, and it is thus not clear if they are guaranteed to hold in the k-wise independent case. We explore the properties of k-wise independent graphs by providing upper-bounds and lower-bounds on the amount of independence, k, required for maintaining the main properties of G(N,p) graphs: connectivity, Hamiltonicity, the connectivity-number, clique-number and chromatic-number and the appearance of fixed subgraphs. Most of these properties are shown to be captured by either constant k or by some k= poly(log(N)) for a wide range of values of p, implying that random looking graphs on N vertices can be generated by a seed of size poly(log(N)). The proofs combine combinatorial, probabilistic and spectral techniques.