Yahav Alon

Motivated by the investigation of sharpness of thresholds for Ramsey properties in random graphs, Friedgut, Kohayakawa, Rödl, Ruciński and Tetali introduced two variants of a single‐player game whose goal is to colour the edges of a random graph, in an online fashion, so as not to create a monochromatic triangle. In the two‐round variant of the game, the player is first asked to find a triangle‐free colouring of the edges of a random graph and then extend this colouring to a triangle‐free colouring of the union of and another (independent) random graph , which is disclosed to the player only after they have coloured . Friedgut et al. analysed this variant of the online Ramsey game in two instances: when has edges and when the number of edges of is just below the threshold above which a random graph typically no longer admits a triangle‐free colouring, which is located at . The two‐round Ramsey game has been recently revisited by Conlon, Das, Lee and Mészáros, who generalised the result of Friedgut et al. from triangles to all strictly 2‐balanced graphs. We extend the work of Friedgut et al. in an orthogonal direction and analyse the triangle case of the two‐round Ramsey game at all intermediate densities. More precisely, for every , with the exception of , we determine the threshold density at which it becomes impossible to extend any triangle‐free colouring of a typical to a triangle‐free colouring of the union of and . An interesting aspect of our result is that this threshold density ‘jumps’ by a polynomial quantity as crosses a “critical” window around .

Let denote the minimum number of edges whose addition to results in a Hamiltonian graph, and let denote the minimum number of edges whose addition to results in a pancyclic graph. We study the distributions of in the context of binomial random graphs. Letting , we prove that there exists a function of order such that, if with , then with high probability . Let denote the number of degree vertices in . A trivial lower bound on is given by the expression . We show that in the random graph process with high probability there exist times , both of order , such that for every and for every . The time can be characterized as the smallest for which contains less than copies of a certain problematic subgraph. In particular, this implies that the hitting time for the existence of a Hamilton path is equal to the hitting time of with high probability. For the binomial random graph, this implies that if and , then, with high probability, . For completion to pancyclicity, we show that if and , then, with high probability, . Finally, we present a polynomial time algorithm such that, if and , then, with high probability, the algorithm returns a set of edges of size whose addition to results in a pancyclic (and therefore also Hamiltonian) graph.