ISSN 1000-1239 CN 11-1777/TP

Journal of Computer Research and Development ›› 2017, Vol. 54 ›› Issue (12): 2843-2850.

### Phase Transition Properties of k-Independent Sets in Random Graphs

Lu Youjun, Xu Daoyun

1. (College of Computer Science and Technology, Guizhou University, Guiyang 550025)
• Online:2017-12-01

Abstract: Phase transition property is one of the most important properties of the theory of Erds-Rényi random graphs. A subset of vertices is a k-independent set in a simple undirected graph G=(V,E) if the subset is an independent set containing k vertex. In order to understand the structural property of k-independent sets in Erds-Rényi random graphs, the phase transition properties of k-independent sets in Erds-Rényi random graphs are investigated in this paper. It is shown that the threshold probability is p\-c=1－n\+－2/k－1 for the existence of k-independent sets in random graph G(n,p) via the first moment method and the second moment method when 2≤k=ο(n). According to this fact that random graph G(n,p) is equivalent to random graph G(n,m) when m is close to pC\+2\-n, the threshold edge number is given by m\-c=［n(n－1)/2(1－n\+－2/k－1)］ for the existence of k-independent sets in random graph G(n,m). The simulation results show that the consistence between simulation and theoretical threshold value for the existence of k-independent sets in random graph G(n,p) and G(n,m) when 2≤k=ο(n), and the threshold value is related to the total number n of vertices and the number k of vertices of independent set. However, when k=ω(n), the theoretical threshold value is not consistent with the simulation threshold value for the existence of k-independent sets in random graph G(n,p) and G(n,m).

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