Sharp thresholds for Hamiltonicity in random intersection graphs

Random Intersection Graphs, G"n","m","p, is a class of random graphs introduced in Karonski (1999) [7] where each of the n vertices chooses independently a random subset of a universal set of m elements. Each element of the universal sets is chosen independently by some vertex with probability p. Two vertices are joined by an edge iff their chosen element sets intersect. Given n, m so that m=@?n^@a@?, for any real @a different than one, we establish here, for the first time, a sharp threshold for the graph property ''Contains a Hamilton cycle''. Our proof involves new, nontrivial, coupling techniques that allow us to circumvent the edge dependencies in the random intersection graph model.

[1]  D. Aldous.  Random walks on finite groups and rapidly mixing markov chains , 1983 .

[2]  Edward R. Scheinerman,et al.  On Random Intersection Graphs: The Subgraph Problem , 1999, Combinatorics, Probability and Computing.

[3]  M. P. Alfaro,et al.  Solution of a problem of P. Tura´n on zeros of orthogonal polynomials on the unit circle , 1988 .

[4]  Paul G. Spirakis,et al.  Expander properties and the cover time of random intersection graphs , 2007, Theor. Comput. Sci..

[5]  János Komlós,et al.  Limit distribution for the existence of Hamiltonian cycles in a random graph , 2006, Discret. Math..

[6]  James Allen Fill,et al.  Random intersection graphs when m= w (n): an equivalence theorem relating the evolution of the G ( n, m, p ) and G ( n,P /italic>) models , 2000 .

[7]  Sheldon M. Ross,et al.  Stochastic Processes , 2018, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.

[8]  Willemien Kets,et al.  RANDOM INTERSECTION GRAPHS WITH TUNABLE DEGREE DISTRIBUTION AND CLUSTERING , 2009, Probability in the Engineering and Informational Sciences.

[9]  Paul G. Spirakis,et al.  Tail bounds for occupancy and the satisfiability threshold conjecture , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[10]  Edward Szpilrajn-Marczewski.  Sur deux propriétés des classes d'ensembles , 1945 .

[11]  James Allen Fill,et al.  Random intersection graphs when m=omega(n): An equivalence theorem relating the evolution of the G(n, m, p) and G(n, p) models , 2000, Random Struct. Algorithms.

[12]  Paul G. Spirakis,et al.  On the Existence of Hamiltonian Cycles in Random Intersection Graphs , 2005, ICALP.

[13]  Josep Díaz,et al.  Sharp threshold for hamiltonicity of random geometric graphs , 2006, SIAM J. Discret. Math..

[14]  Paul G. Spirakis,et al.  Simple and Efficient Greedy Algorithms for Hamilton Cycles in Random Intersection Graphs , 2005, ISAAC.

[15]  Béla Bollobás,et al.  Random Graphs , 1985 .

[16]  Paul G. Spirakis,et al.  The Existence and Efficient Construction of Large Independent Sets in General Random Intersection Graphs , 2004, ICALP.

[17]  Dudley Stark.  The vertex degree distribution of random intersection graphs , 2004 .

[18]  Brendan D. McKay,et al.  The degree sequence of a random graph. I. The models , 1997 .

[19]  Brendan D. McKay,et al.  The degree sequence of a random graph. I. The models , 1997, Random Struct. Algorithms.