Hamilton cycles in the semi-random graph process

The semi-random graph process is a single player game in which the initially presented an empty on n vertices. In each round, vertex u to independently and uniformly at random. then adaptively selects v, adds edge uv graph. For fixed monotone property, objective of force satisfy this property with high probability as few rounds possible. We focus problem constructing Hamilton cycle particular, we present novel strategy for achieves Hamiltonian c∗n rounds, where value c∗ result dimensional optimization problem. Numerical computations indicate that c∗<2.61135. This improves upon previously best known upper bound 3n rounds. also show lower (ln2+ln(1+ln2)+o(1))n not tight.