Resilience of graphs

In this paper, we initiate a systematic study of graph resilience. The (local) resilience of a graph G with respect to a property P measures how much one has to change G (locally) in order to destroy P . Estimating the resilience leads to many new and challenging problems. Here we focus on random and pseudo-random graphs and prove several sharp results. 1 The notion of resilience A typical result in graph theory is of the following form A graph G (from certain class) possesses a property P. In this paper, we would like to investigate the following general problem How strongly does G posses P? To study this question, we define the resilience of G with respect to P, which measures how much one should change G in order to destroy P. There are two natural kinds of resilience: global and local. It is more convenient to first define these quantities with respect to monotone increasing properties (P is monotone increasing if it is preserved under edge addition). Definition 1.1 (Global resilience) Let P be an increasing monotone property. The global resilience of G with respect to P is the minimum number r such that one can destroy P by deleting r edges from G. The notion of global resilience is not new. In fact, problems about global resilience are popular in extremal graph theory. For example, the celebrated theorem of Turán [17] gives the answer to the following question. How many edges should one delete from the complete graph Kn to make it Kk-free ? The main focus of this paper is on the local resilience , which eventually leads to a host of intriguing new questions. To start, let us notice that one can destroy many properties by simple local changes. For instance, to destroy the hamiltonicity, it suffices to delete all edges adjacent to one vertex. This motivates the following notion. Department of Mathematics, Princeton University, Princeton, NJ 08544, and Institute for Advanced Study, Princeton. E-mail: [email protected]. Research supported in part by NSF CAREER award DMS-0546523, NSF grant DMS-0355497, USA-Israeli BSF grant, Alfred P. Sloan fellowship, and the State of New Jersey. Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA. E-mail: [email protected]. Research supported in part by by NSF CAREER award and by an Alfred P. Sloan fellowship.