Roman Glebov

On extremal hypergraphs for Hamiltonian cycles

We study sufficient conditions for Hamiltonian cycles in hypergraphs, and obtain both Turan- and Dirac-type results. While the Turan-type result gives an exact threshold for the appearance of a Hamiltonian cycle in a hypergraph depending only on the extremal number of a certain path, the Dirac-type result yields a sufficient condition relying solely on the minimum vertex degree.

On the Number of Hamilton Cycles in Sparse Random Graphs

We prove that the number of Hamilton cycles in the random graph

On the maximum number of Latin transversals

G(n,p)

On the Concentration of the Domination Number of the Random Graph

is

On covering expander graphs by hamilton cycles

n!p^n(1+o(1))^n

Building spanning trees quickly in Maker-Breaker games

asymptotically almost surely (a.a.s.), provided that

Biased games on random boards

p\geq \frac{\ln n+\ln\ln n+\omega(1)}{n}

The number of Hamiltonian decompositions of regular graphs

. Furthermore, we prove the hitting time version of this statement, showing that in the random graph process, the edge that creates a graph of minimum degree

The Threshold Probability for Long Cycles

2

Conflict-free colouring of graphs

creates