The degree sequence of a random graph. I. The models
Random Structures & Algorithms
On Random Intersection Graphs: The Subgraph Problem
Combinatorics, Probability and Computing
Expander properties and the cover time of random intersection graphs
Theoretical Computer Science
The second eigenvalue of random walks on symmetric random intersection graphs
CAI'07 Proceedings of the 2nd international conference on Algebraic informatics
Sharp thresholds for Hamiltonicity in random intersection graphs
Theoretical Computer Science
On the independence number and Hamiltonicity of uniform random intersection graphs
Theoretical Computer Science
Simple and efficient greedy algorithms for hamilton cycles in random intersection graphs
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
Selected combinatorial properties of random intersection graphs
Algebraic Foundations in Computer Science
Expander properties and the cover time of random intersection graphs
MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
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Random Intersection Graphs is a new class of random graphs introduced in [5], in which each of n vertices randomly and independently chooses some elements from a universal set, of cardinality m. Each element is chosen with probability p. Two vertices are joined by an edge iff their chosen element sets intersect. Given n, m so that m=nα, for any real α different than one, we establish here, for the first time, tight lower bounds p0(n,m), on the value of p, as a function of n and m, above which the graph Gn,m,p is almost certainly Hamiltonian, i.e. it contains a Hamilton Cycle almost certainly. Our bounds are tight in the sense that when p is asymptotically smaller than p0(n,m) then Gn,m,p almost surely has a vertex of degree less than 2. Our proof involves new, nontrivial, coupling techniques that allow us to circumvent the edge dependencies in the random intersection model. Interestingly, Hamiltonicity appears well below the general thresholds, of [4], at which Gn,m,p looks like a usual random graph. Thus our bounds are much stronger than the trivial bounds implied by those thresholds. Our results strongly support the existence of a threshold for Hamiltonicity in Gn,m,p.