We prove that the ‘Upper Matching Conjecture’ of Friedland, Krop, and Markström and the analogous conjecture of Kahn for independent sets in regular graphs hold for all large enough graphs as a function of the degree. That is, for every d and every large enough n divisible by 2d, a union of $n/(2d)$ copies of the complete d-regular bipartite graph maximizes the number of independent sets and matchings of size k for each k over all d-regular graphs on n vertices. To prove this we utilize the cluster expansion for the canonical ensemble of a statistical physics spin model, and we give some further applications of this method to maximizing and minimizing the number of independent sets and matchings of a given size in regular graphs of a given minimum girth.

我们证明了 Friedland、Krop 和 Markström 的“上匹配猜想”以及 Kahn 对正则图中独立集的类似猜想对于所有足够大的图作为度的函数都成立。也就是说，对于每个d和每个足够大的n可被 2 d整除，$n/(2d)$完整的拷贝d -regular二分图最大化独立集和大小的匹配数的数量ķ每个ķ在所有d上-regular图表Ñ顶点。为了证明这一点，我们将集群扩展用于统计物理自旋模型的规范系综，并且我们给出了这种方法的一些进一步应用，以最大化和最小化给定最小值的正则图中给定大小的独立集和匹配的数量周长。