Random Graphs

Abstract

For random graphs, the following results were shown in Gu et al. (Theor Comput Sci 609:336–343, 2016). Here let G(n, p) denote the Erdős-Renyi (Erdős and Rényi, Magy Tud Akad Mat Kutató Int Közl 5:17–61, 1960) random graph with n vertices and edges appearing with probability p. We say an event $\mathcal {A}$ happens with high probability if the probability that it happens approaches 1 as n →∞, i.e., $Pr[\mathcal {A}]=1-o_n(1)$. Sometimes, we say w.h.p. for short. We say that a property holds for almost all graphs if the probability of the property holding for G(n, 1∕2) approaches 1 as n approaches infinity. The first result follows easily from Theorem 3.0.2 and the fact that almost all graphs are 3-connected (Blass and Harary, J Graph Theory 3:225–240, 1979).