Topics in Extremal and Probabilistic Combinatorics

Abstract

In this thesis, we consider a collection of problems in extremal and probabilistic combinatorics, specifically graph theory. First, we consider a close relative of the Cops and Robbers game called Revolutionaries and Spies, a two-player pursuit/evasion game devised by Beck to model network security. We show that on a ‘typical’ graph, if the second player has fewer pieces than are required to execute a particular trivial winning strategy, then the game is a first player win. Second, we consider the emergence of the square of a Hamilton cycle in a random geometric graph process, and show that typically, the exact instant at which a simple local obstacle is eliminated at every vertex, is the exact instant at which the graph becomes square Hamiltonian. This is in stark contrast to the ‘normal’ Erdos-Rényi random graph process, in which square Hamiltonicity is both not ‘local' in this sense, and occurs only once the graph is reasonably dense. Finally, we study an extremal problem concerning tournaments, that of maximising the number of oriented cycles of a fixed length. A ‘folklore’ result states that for 3-cycles one cannot do significantly better than a random tournament. More recent work shows that same is true for 5-cycles, and perhaps surprisingly that this is not true for 4-cycles. We conjecture that one can significantly beat the random tournament in expectation if and only if the length of the cycle is divisible by four, proving the ‘if’ statement, as well as a variety of new cases of the ‘only if' statement, including the case that the graph is sufficiently close to being regular.