This article extend the John E. Hopcroft and Richart M. Karp Algorithm (HK Algorithm) for maximum matchings in bipartite graphs to the non-bipartite case by providing a new approach to deal with the blossom in alternating paths in the process of searching for augmenting paths, which different from well-known "shrinking" way of Edmonds and makes the algorithm for maximum matchings in general graphs more simple.

In this paper we consider the Hamilton cycle problem in the rectangular meshes with at most two faulty nodes. We prove that this problem is solvable in polynomial time with a corresponding algorithm. We provided an entirely new approach to this problem being different from the method early used on this problem.

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