Giant Component and Poisson Cloning Model

The notion of a random graph was first introduced in 1947 by Erdös [27] to show the existence of a graph with a certain Ramsey property. A decade later, the theory of the random graph began with the paper entitled “On Random Graphs I” by Erdös and R、 enyi [28], and the theory had been developed by a series [29, 30, 31, 32, 33] of their papers. Since then, the subject has become one of the most active research areas. Many researchers have devoted themselves to studying various properties of random graphs, such as the emergence of the giant component [29, 9, 53], the connectivity [28, 30, 16], the existence of perfect matching [31, 32, 33, 16], the existence of Hamiltonian cycle(s) [52, 10, 15], the core problem [10, 55, 62], and the graph invariants like the independence number [14, 57] and the chromatic number [63, 12, 54]. (The list of references here is far from being exhaustive.) There are two canonical models for random graphs, both of which were originated in the simple model introduced in [27]. In the binomial model G(n, p) on a set V of n vertices, each of a k possible edges is in the graph with probability p, independently of other edges. Thus, the probability of G(n, p) being a fixed graph G with m edges is pm(1 p) j-. The uniform model G(n, m) on V is a graph chosen uniformly at random from the set of all graphs on V with m edges. Hence, G(n, m) becomes a fixed graph G with probability am )k-1, provided G has m edges. Most of asymptotic behaviors of the two models are almost identical if their expected numbers of edges are the same. (See Proposition 1.13 in [45].) The random graph process, in which random edges are added one by one, is also extensively studied. For more about models and/or basics of random graphs, we recommend two books with the same title Random Graphs by Bollobás [11], and by Janson, Luczak and Ruci、 nski [45]. The phase transition phenomenon is among interesting topics of random graphs. Specifically, the phase transition phenomena regarding the emergences of the giant (connected) component and the t-core problem have attracted much attention. In their monumental paper entitled “On the Evolution of Random Graphs” [29], Erdös and R、 enyi proved that, for the size 1(n, p) of a largest component of G(n, p), n 2 n 2 n 2 Giant Component and Poisson Cloning Model