We study the localization game on dense random graphs. In this game, a cop x tries to locate a robber y by asking for the graph distance of y from every vertex in a sequence of sets W1,W2, . . . ,W`. We prove high probability upper and lower bounds for the minimum size of each Wi that will guarantee that x will be able to locate y.

We consider a multitype epidemic model which is a natural extension of the randomized Reed–Frost epidemic model. The main result is the derivation of an asymptotic Gaussian limit theorem for the final size of the epidemic. The method of proof is simpler, and more direct, than is used for similar results elsewhere in the epidemics literature. In particular, the results are specialized to epidemi...

Random walks on random graphs – p. 1/2

We consider a multitype epidemic model which is a natural extension of the randomised Reed-Frost epidemic model. The main result is the derivation of an asympotic Gaussian limit theorem for the final size of the epidemic. The method of proof is simpler, and more direct, than is used for similar results elsewhere in the epidemics literature. In particular, the results are specialised to epidemic...

We provide sufficient conditions for the existence of long cycles in locally expanding graphs, and present applications of our conditions and techniques to Ramsey theory, random graphs and positional games.