A Polynomial time Algorithm for Hamilton Cycle with maximum Degree 3

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

Hamilton cycles in hypergraphs below the Dirac threshold

We establish a precise characterisation of $4$-uniform hypergraphs with minimum codegree close to $n/2$ which contain a Hamilton $2$-cycle. As an immediate corollary we identify the exact Dirac threshold for Hamilton $2$-cycles in $4$-uniform hypergraphs. Moreover, by derandomising the proof of our characterisation we provide a polynomial-time algorithm which, given a $4$-uniform hypergraph $H$ with minimum codegree close to $n/2$, either finds a Hamilton $2$-cycle in $H$ or provides a certificate that no such cycle exists. This surprising result stands in contrast to the graph setting, in which below the Dirac threshold it is NP-hard to determine if a graph is Hamiltonian. We also consider tight Hamilton cycles in $k$-uniform hypergraphs $H$ for $k \geq 3$, giving a series of reductions to show that it is NP-hard to determine whether a $k$-uniform hypergraph $H$ with minimum degree $\delta(H) \geq \frac{1}{2}|V(H)| - O(1)$ contains a tight Hamilton cycle. It is therefore unlikely that a similar characterisation can be obtained for tight Hamilton cycles.Comment: v2: minor revisions in response to reviewer comments, most pseudocode and details of the polynomial time reduction moved to the appendix which will not appear in the printed version of the paper. To appear in Journal of Combinatorial Theory, Series

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

Determinant Sums for Undirected Hamiltonicity

We present a Monte Carlo algorithm for Hamiltonicity detection in an $n$-vertex undirected graph running in $O^*(1.657^{n})$ time. To the best of our knowledge, this is the first superpolynomial improvement on the worst case runtime for the problem since the $O^*(2^n)$ bound established for TSP almost fifty years ago (Bellman 1962, Held and Karp 1962). It answers in part the first open problem in Woeginger's 2003 survey on exact algorithms for NP-hard problems. For bipartite graphs, we improve the bound to $O^*(1.414^{n})$ time. Both the bipartite and the general algorithm can be implemented to use space polynomial in $n$. We combine several recently resurrected ideas to get the results. Our main technical contribution is a new reduction inspired by the algebraic sieving method for $k$-Path (Koutis ICALP 2008, Williams IPL 2009). We introduce the Labeled Cycle Cover Sum in which we are set to count weighted arc labeled cycle covers over a finite field of characteristic two. We reduce Hamiltonicity to Labeled Cycle Cover Sum and apply the determinant summation technique for Exact Set Covers (Bj\"orklund STACS 2010) to evaluate it.Comment: To appear at IEEE FOCS 201