On the independence number and Hamiltonicity of uniform random intersection graphs

On the independence number and Hamiltonicity of uniform random intersection graphs

In the uniform random intersection graphs model, denoted by G n,m,λ , to each vertex v we assign exactly λ randomly chosen labels of some label set M of m labels and we connect every pair of vertices that has at least one label in common. In this model, we estimate the independence number α(G n,m,λ), for the wide, interesting range m = n α , α < 1 and λ = O(m 1/4). We also prove the hamiltonici...

On the independence number of random graphs

Hamiltonicity, independence number, and pancyclicity

A graph on n vertices is called pancyclic if it contains a cycle of length ` for all 3 ≤ ` ≤ n. In 1972, Erdős proved that if G is a Hamiltonian graph on n > 4k vertices with independence number k, then G is pancyclic. He then suggested that n = Ω(k) should already be enough to guarantee pancyclicity. Improving on his and some other later results, we prove that there exists a constant c such th...

On the b-Independence Number of Sparse Random Graphs

Let graph G = (V,E) and integer b ≥ 1 be given. A set S ⊆ V is said to be b-independent if u, v ∈ S implies dG(u, v) > b where dG(u, v) is the shortest distance between u and v in G. The b-independence number αb(G) is the size of the largest b-independent subset of G. When b = 1 this reduces to the standard definition of independence number. We study this parameter in relation to the random gra...

On The Independence Number Of Random Interval Graphs

A random interval graph of order n is generated by picking 2n numbers X 1 : : : X 2n independently from the uniform distribution on 0; 1] and considering the collection of n intervals with extremities X 2i?1 and X 2i for i 2 f1;:::ng. The graph vertices correspond to intervals. Two vertices are connected if the corresponding intervals intersect. This paper characterizes the uctuations of the in...