Infinite paths and cliques in random graphs

Infinite Paths in Random Shift Graphs

We determine the probability tresholds for the existence of infinite paths in random shift graphs.

Cliques in random graphs

1. Introduction. Let 0 < p < 1 be fixed and denote by G a random graph with point set N, the set of natural numbers, such that each edge occurs with probability p, independently of all other edges. In other words the random variables ei5 , 1 < i < j, defined by _ 1 if (i, j) is an edge of G, et '-0 if (i, j) is not an edge of G, are independent r .v.'s with P(e i, = 1) = p, P(eij = 0) = 1-p. De...

Infinite paths in planar graphs III, 1-way infinite paths

An infinite graph is 2-indivisible if the deletion of any finite set of vertices from the graph results in exactly one infinite component. Let G be a 4-connected, 2indivisible, infinite, plane graph. It is known that G contains a spanning 1-way infinite path. In this paper, we prove a stronger result by showing that, for any vertex x and any edge e on a facial cycle of G, there is a spanning 1-...

Cliques in dense inhomogeneous random graphs

Large Cliques in Sparse Random Intersection Graphs

Given positive integers n and m, and a probability measure P on {0, 1, . . . ,m}, the random intersection graph G(n,m,P ) on vertex set V = {1, 2, . . . , n} and with attribute set W = {w1, w2, . . . , wm} is defined as follows. Let S1, S2, . . . , Sn be independent random subsets of W such that for any v ∈ V and any S ⊆ W we have P(Sv = S) = P (|S|)/ ( m |S| ) . The edge set of G(n,m,P ) consi...