A Polynomial time Algorithm for Hamilton Cycle with maximum Degree 3

Abstract

Based on the famous Rotation-Extension technique, by creating the new concepts and methods: broad cycle, main segment, useful cut and insert, destroying edges for a main segment, main goal Hamilton cycle, depth-first search tree, we develop a polynomial time algorithm for a famous NPC: the Hamilton cycle problem. Thus we proved that NP=P. The key points of this paper are: 1) there are two ways to get a Hamilton cycle in exponential time: a full permutation of n vertices; or, chose n edges from all k edges, and check all possible combinations. The main problem is: how to avoid checking all combinations of n edges from all edges. My algorithm can avoid this. Lemma 1 and lemma 2 are very important. They are the foundation that we always can get a good branch in the depth-first search tree and can get a series of destroying edges (all are bad edges) for this good branch in polynomial time. The extraordinary insights are: destroying edges, a tree contains each main segment at most one time at the same time, and dynamic combinations. The difficult part is to understand how to construct a main segment's series of destroying edges by dynamic combinations (see the proof of lemma 4). The proof logic is: if there is at least on Hamilton cycle in the graph, we always can do useful cut and inserts until a Hamilton cycle is got. The times of useful cut and inserts are polynomial. So if at any step we cannot have a useful cut and insert, this means that there are no Hamilton cycles in the graph.Comment: 49 pages. This time, I add a detailed polynomial time algorithm and proof for 3S