Properties of random coverings of graphs

In the thesis we study selected properties of random coverings of graphs introduced by Amit and Linial in 2002. A random n-covering of a graph G, denoted by G̃, is obtained by replacing each vertex v of G by an n-element set G̃v and then choosing, independently for every edge e = {x, y} ∈ E(G), uniformly at random a perfect matching between G̃x and G̃y. The first problem we consider is the typical size of the largest topological clique in a random covering of given graph G. We show that asymptotically almost surely a random n-covering G̃ of a graph G contains the largest topological clique which is allowed by the structure of G. The second property we examine is the existence of a Hamilton cycle in G̃. We show that if G has minimum degree at least 5 and contains two edge disjoint Hamilton cycles whose union is not a bipartite graph, then asymptotically almost surely G̃ is Hamiltonian.