Threshold functions for small subgraphs in simple graphs and multigraphs

Threshold functions for small subgraphs: an analytic approach

We revisit the problem of counting the number of copies of a fixed graph in a random graph or multigraph, including the case of constrained degrees. Our approach relies heavily on analytic combinatorics and on the notion of patchwork to describe the possible overlapping of copies. This paper is a version, extended to include proofs, of the paper with the same title to be presented at the Euroco...

On the threshold for k-regular subgraphs of random graphs

The k-core of a graph is the largest subgraph of minimum degree at least k. We show that for k sufficiently large, the (k + 2)-core of a random graph G(n, p) asymptotically almost surely has a spanning k-regular subgraph. Thus the threshold for the appearance of a k-regular subgraph of a random graph is at most the threshold for the (k + 2)-core. In particular, this pins down the point of appea...

On small subgraphs of random graphs

Small subgraphs of random regular graphs

The main aim of this short paper is to answer the following question. Given a fixed graph H , for which values of the degree d does a random d-regular graph on n vertices contain a copy of H with probability close to one?

Spanning 3-colourable subgraphs of small bandwidth in dense graphs

A conjecture by Bollobás and Komlós states the following: For every γ > 0 and integers r ≥ 2 and ∆, there exists β > 0 with the following property. If G is a sufficiently large graph with n vertices and minimum degree at least ( r−1 r + γ)n and H is an r-chromatic graph with n vertices, bandwidth at most βn and maximum degree at most ∆, then G contains a copy of H. This conjecture generalises s...