… We consider the random graph model %(n, p); that is, the probability space of simple graphs
… - p’) 2 8(log n)4/ n, then the (orientable) genus of a random graph in %(a, p) is (1 + o(l))pn2/…

… Determining the genus of a graph is a fundamental yet very … that the orientable genus of a
randomly selected graph on n … lower bounds on the genus which hold for almost all graphs. …

… graph allows for both loops and multiple edges. The random graph, on the other hand, is a
simple graph … consists of all simple (or simplicial) graphs on a fixed set of labeled vertices. …

… The Euler genus eg(G) of a connected graph G is the least Euler genus of an embedding.
Then eg(G) ⩾ eg(H ) for any connected subgraph H of G, and eg(G) is the sum of the Euler …

… definitions and properties in topological graph theory and discuss random graphs. Also, our
… D ofa random graph G is a family of digraphs defined on the same vertex set of graphs in G, …

… graph R on the same vertex set, and we consider the genus g(G) of the … recent work
concerning random graphs on given surfaces. The typical properties of graphs with genus at most g …

… ) is to consider random graphs on other surfaces. Here, we define an orientable surface of
genus g as a sphere to which g handles have been attached, and the genus of a graph as the …

… Much of this book concentrates on geometric properties of the random graphs: primarily …
taking place on these graphs, which is one of the two meanings of our title, Random Graph …

… graphs,” ie subgraphs whose removal leaves a planar graph. In particular, we give optimal
bounds for random … present exponentially improved dependence on genus for a number of …

… We investigate Voronoi-like tessellations of bipartite quadrangulations on surfaces of arbitrary
genus, by using a natural generalization of a bijection of Marcus and Schaeffer allowing …