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Based on the famous Rotation-Extension technique, by creating the newconcepts and methods: broad cycle, main segment, useful cut and insert,destroying edges for a main segment, main goal Hamilton cycle, depth-firstsearch tree, we develop a polynomial time algorithm for a famous NPC: theHamilton cycle problem. Thus we proved that NP=P. The key points of this paperare: 1) there are two ways to get a Hamilton cycle in exponential time: a fullpermutation of n vertices; or, chose n edges from all k edges, and check allpossible combinations. The main problem is: how to avoid checking allcombinations of n edges from all edges. My algorithm can avoid this. Lemma 1and lemma 2 are very important. They are the foundation that we always can geta good branch in the depth-first search tree and can get a series of destroyingedges (all are bad edges) for this good branch in polynomial time. Theextraordinary insights are: destroying edges, a tree contains each main segmentat most one time at the same time, and dynamic combinations. The difficult partis to understand how to construct a main segment's series of destroying edgesby dynamic combinations. The proof logic is: if there is at least on Hamiltoncycle in the graph, we always can do useful cut and inserts until a Hamiltoncycle is got. The times of useful cut and inserts are polynomial. So if at anystep we cannot have a useful cut and insert, this means that there are noHamilton cycles in the graph. In this version, I add a detailed polynomial timealgorithm and proof for 3SAT

A Polynomial time Algorithm for Hamilton Cycle with maximum Degree 3, 3SAT