稠密图的连通集
Contagious Sets in Dense Graphs
摘要：
我们研究了无向图中的激活过程，称为自举渗透：一个顶点是活跃的，如果它属于一组初始激活的顶点，或者如果在某个点上，它至少有r个活跃的邻居，对于一个阈值r，这是相同的所有顶点。传染集是一个顶点集，其激活结果是整个图都是活动的。设m（G，r）是n阶图G中最小传染集的大小.我们研究密度条件，确保m（G，r）= r的所有r >= 2。关于最小度，我们证明了保证存在这样一个最小可能传染集当且仅当G的最小度至少为（k-1）/k * n.此外，我们研究了激活传播的速度，并提供了严格的上界，直到所有节点都被激活，在这样的图轮数。我们还调查了什么平均度断言存在小的传染集。对于n >= k >= r，我们用M（n，k，r）表示满足m（G，r）>k的n-顶点图G的最大边数.我们确定M（n，k，2）和M（n，k，k）的精确值，假设n与k相比足够大。
英文摘要：
We study the activation process in undirected graphs known as bootstrap percolation: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it had at least r active neighbors, for a threshold r that is identical for all vertices. A contagious set is a vertex set whose activation results with the entire graph being active. Let m(G,r) be the size of a smallest contagious set in a graph G on n vertices. We examine density conditions that ensure m(G,r) = r for all r >= 2. With respect to the minimum degree, we prove that such a smallest possible contagious set is guaranteed to exist if and only if G has minimum degree at least (k-1)/k * n. Moreover, we study the speed with which the activation spreads and provide tight upper bounds on the number of rounds it takes until all nodes are activated in such a graph. We also investigate what average degree asserts the existence of small contagious sets. For n >= k >= r, we denote by M(n,k,r) the maximum number of edges in an n-vertex graph G satisfying m(G,r)>k. We determine the precise value of M(n,k,2) and M(n,k,k), assuming that n is sufficiently large compared to k.
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