We consider percolation on high‐dimensional product graphs, where the base graphs are
regular and of bounded order. In the subcritical regime, we show that typically the largest …

We introduce a notion of the crux of a graph G, measuring the order of a smallest dense
subgraph in G. This simple-looking notion leads to some generalizations of known results …

We study Hamiltonicity in random subgraphs of the hypercube $\mathcal {Q}^ n $. Our first
main theorem is an optimal hitting time result. Consider the random process which includes …

Carsten Thomassen in 1989 conjectured that if a graph has minimum degree much more
than the number of atoms in the universe (δ (G)≥ 10 10 10), then it contains a pillar, which is …

It is known that many different types of finite random subgraph models undergo
quantitatively similar phase transitions around their percolation thresholds, and the proofs of …

In their seminal paper introducing the theory of random graphs, Erd\H {o} s and R\'{e} nyi
considered the evolution of the structure of a random subgraph of $ K_n $ as the density …

Carsten Thomassen in 1989 conjectured that if a graph has minimum degree more than the
number of atoms in the universe (δ (G)≥ 101010), then it contains a pillar, which is a graph …

We study regular graphs in which the random walks starting from a positive fraction of
vertices have small mixing time. We prove that any such graph is virtually an expander and …

In 2004, Frieze, Krivelevich and Martin established the emergence of a giant component in
random subgraphs of pseudo‐random graphs. We study several typical properties of the …

It is known that many different types of finite random subgraph models undergo
quantitatively similar phase transitions around their percolation thresholds, and the proofs of …