Filling random cycles

On Cycles in Random Graphs

We consider the geometric random (GR) graph on the d−dimensional torus with the Lσ distance measure (1 ≤ σ ≤ ∞). Our main result is an exact characterization of the probability that a particular labeled cycle exists in this random graph. For σ = 2 and σ = ∞, we use this characterization to derive a series which evaluates to the cycle probability. We thus obtain an exact formula for the expected...

Hamilton cycles in random subgraphs of pseudo-random graphs

Packing hamilton cycles in random and pseudo-random hypergraphs

We say that a k-uniform hypergraph C is a Hamilton cycle of type `, for some 1 ≤ ` ≤ k, if there exists a cyclic ordering of the vertices of C such that every edge consists of k consecutive vertices and for every pair of consecutive edges Ei−1, Ei in C (in the natural ordering of the edges) we have |Ei−1 \ Ei| = `. We prove that for k/2 < ` ≤ k, with high probability almost all edges of the ran...

Longest cycles in sparse random digraphs

Long paths and cycles in sparse random graphs and digraphs were studied intensively in the 1980’s. It was finally shown by Frieze in 1986 that the random graph G(n, p) with p = c/n has a cycle on at all but at most (1 + ε)ce−cn vertices with high probability, where ε = ε(c) → 0 as c → ∞. This estimate on the number of uncovered vertices is essentially tight due to vertices of degree 1. However,...

Colouring powers of cycles from random lists

Let C n be the k-th power of a cycle on n vertices (i.e. the vertices of C n are those of the n-cycle, and two vertices are connected by an edge if their distance along the cycle is at most k). For each vertex draw uniformly at random a subset of size c from a base set S of size s = s(n). In this paper we solve the problem of determining the asymptotic probability of the existence of a proper c...