Inclusion and exclusion algorithm for the Hamiltonian path problem

Hamiltonian cycles and paths in vertex-transitive graphs with abelian and nilpotent groups

The n-dimensional locally twisted cube LTQn, a variation of the hypercube Qn, has the same number of vertices and the same number of edges as Qn, but it has only about half of the diameter of Qn. The existence of the Hamiltonian cycle provides an advantage when implementing algorithms that require a ring structure. In addition, k (≥ 2) edge-disjoint Hamiltonian cycles also provide the edge-fault tolerant Hamiltonicity for the interconnection network. Hung [Theoretical Computer Science 412, 4747–4753, 2011] proved that LTQn with n ⩾ 4 contains two edge-disjoint Hamiltonian cycles, and posted an open problem what is the maximum number of edge-disjoint Hamiltonian cycles in LTQn for n ⩾ 6? In this paper, we show that there exist three edge-disjoint Hamiltonian cycles on LTQn while n ⩾ 6.

This paper shows that graph traversal techniques have fundamental differences between serial and distributed computations in their behaviors, computational complexities, and effects on the design of graph algorithms. It has three major parts. Section I describes the computational environment for the design and description of distributed graph algorithms in terms of an architectural model for message exchanges. The computational complexity is measured in terms of the number of messages transmitted. Section II presents several distributed algorithms for the pure traversal, depth-first search, and breadth-first search techniques. Their complexities are also given. Through these descriptions are brought out some of the intrinsic differences in the behaviors and complexities of the fundamental traversal techniques between a serial and a distributed computation environment. Section III gives the distributed version of the Ford and Fulkerson algorithm for the maximum flow problem by means of depth-first search, the largest-augmentation search and breadth-first search. The complexities of these methods are found to be 0(f*|A|), 0((l + logM/(M-1)f*|V||A|) and O(|V|6), respectively, where f* is the maximum flow value of the problem, M is the maximum number of ucs in a cut, |V| is the number of vertices, and |A| is the number of arcs. Lastly, it is shown that the largest augmentation search may be a better method than the other two. This is contrary to the known results in serial computation.

The Stirling numbers of the first and second kind enumerate the permutations of a finite set that are decomposed into a given number of cycles and the partitions of a finite set into a given number of subsets, respectively. Basic properties of these numbers and their generalizations of probabilistic and statistical interest are reviewed. Specifically, explicit expressions, generating functions, recurrence relations, and other useful properties of these numbers are presented. Further, the Lah numbers and the generalized factorial coefficients, closely related to the Stirling numbers, are discussed. The noncentral and the associated Stirling and related numbers are briefly presented. The generalized Stirling and Lah numbers and, in particular, the q ‐Stirling and q ‐Lah numbers are only quoted. Probabilistic and statistical applications of the Stirling and related numbers in expressing the probability mass function of certain (a) urn‐model distributions, (b) occupancy distributions, (c) convolutions of truncated discrete distributions and (d) compound discrete distributions are exemplified.

A graph is called Hamiltonian connected if it contains a Hamiltonian path between any two distinct vertices. In the past, we proved the Hamiltonian path and cycle problems for general supergrid graphs to be NP-complete. However, they are still open for solid supergrid graphs. In this paper, first we will verify the Hamiltonian cycle property of C-shaped supergrid graphs, which are a special case of solid supergrid graphs. Next, we show that C-shaped supergrid graphs are Hamiltonian connected except in a few conditions. For these excluding conditions of Hamiltonian connectivity, we compute their longest paths. Then, we design a linear-time algorithm to solve the longest path problem in these graphs. The Hamiltonian connectivity of C-shaped supergrid graphs can be applied to compute the optimal stitching trace of computer embroidery machines, and construct the minimum printing trace of 3D printers with a C-like component being printed.

A note on the middle levels problem

A digraph is semicomplete if it has no pair of non-adjacent vertices. It is complete if every pair of distinct vertices induces a 2-cycle. A digraph is semicomplete multipartite if it can be obtained from a semicomplete digraph D by choosing a collection of vertex-disjoint subsets X₁,…,X_c of V(D) and then deleting all arcs both of whose end-vertices lie inside some X_i. We can also think of a semicomplete digraph as being obtained from a semicomplete multipartite digraph on the same vertex set and partite sets V₁,…,V_c by adding the arcs of a semicomplete digraph D_i on V_i for each partite set V_i. It is well known that both the hamiltonian path and the hamiltonian cycle problem can be solved in polynomial time for semicomplete multipartite digraphs. In this paper we study the complexity of finding a hamiltonian path or cycle in a semicomplete digraph S which is obtained as above from a semicomplete multipartite digraph D and semicomplete digraphs D_i=(V_i,A_i), i∈[c] such that the path or cycle uses as few arcs of A₁∪…A_c as possible. We obtain a number of results for the case when each D_i is a complete digraph. Already this case is highly nontrivial in the cycle case and the complexity is still open. We show how to find a Hamiltonian path which uses as few arcs from the D_i’s as possible in polynomial time and obtain a number of results, both structural and algorithmic on hamiltonian cycles that use the minimum or close to the minimum number of arcs from the D_i’s. Our results imply the polynomial solvability of some special cases of the NP-complete {0,1}-TSP problem. Finally we show that two natural questions about properties of quasi-hamiltonian cycles, that is, cycles meeting all partite sets in semicomplete multipartite digraphs are NP-complete.

In previous work, Feldmann and Mysliwietz [3] showed that every shuffle-exchange graph contains a Hamilton path. In this paper, we show that there is a linear time ranking algorithm associated with that Hamilton path. The ranking algorithm returns the rank-position of each node of the graph with respect to the order given by the Hamilton path. Ranking algorithms are important for they can be applied to yield efficient implementations of certain network emulations using SIMD-style parallel algorithms for translating node labels.KeywordsCayley GraphHamilton PathRanking AlgorithmInitial NodeGray CodeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this article we consider the Directed Steiner Path Cover problem on directed co-graphs. Given a directed graph G=(V,E) and a set T subseteq V of so-called terminal vertices, the problem is to find a minimum number of vertex-disjoint simple directed paths, which contain all terminal vertices and a minimum number of non-terminal vertices (Steiner vertices). The primary minimization criteria is the number of paths. We show how to compute in linear time a minimum Steiner path cover for directed co-graphs. This leads to a linear time computation of an optimal directed Steiner path on directed co-graphs, if it exists. Since the Steiner path problem generalizes the Hamiltonian path problem, our results imply the first linear time algorithm for the directed Hamiltonian path problem on directed co-graphs. We also give binary integer programs for the (directed) Hamiltonian path problem, for the (directed) Steiner path problem, and for the (directed) Steiner path cover problem. These integer programs can be used to minimize change-over times in pick-and-place machines used by companies in electronic industry.

An efficient parallel algorithm for testing whether a graph G is k-vertex connected is presented. The algorithm runs in $O(k^2 \log n)$ time and uses $(n + k^2 )kC(n,m)$ processors on a CROW PRAM, where n and m are the number of vertices and edges of G, and $C(n,m)$ is the number of processors required to compute the connected components of G in logarithmic time. For fixed k, the algorithm runs in logarithmic time and uses $nC(n,m)$ processors. To develop our algorithm, an efficient parallel algorithm is designed for the following disjoint s-t paths problem. Given a graph G, and two specified vertices s and t, find k vertex disjoint paths between s and t, if they exist. If no such paths exist, find a set of at most $k-1$ vertices whose removal disconnects s and t. The parallel algorithm for this problem runs in $O(k^2 \log n)$ time and uses $kC(n,m)$ processors. The way to modify the algorithm to find k-edge disjoint paths, if they exist, is shown. This yields an efficient parallel algorithm for testing w...

Not being (super)thin or solid is hard: A study of grid Hamiltonicity

We consider the complexity of the Hamilton cycle decision problem when restricted to k-uniform hypergraphs H of high minimum codegree delta(H). We show that for tight Hamilton cycles this problem is NP-hard even when restricted to k-uniform hypergraphs H with delta(H) >= n/2 - C, where n is the order of H and C is a constant which depends only on k. This answers a question raised by Karpinski, Rucinski and Szymanska. Additionally we give a polynomial-time algorithm which, for a sufficiently small constant epsilon > 0, determines whether or not a 4-uniform hypergraph H on n vertices with delta(H) >= n/2 - epsilon * n contains a Hamilton 2-cycle. This demonstrates that some looser Hamilton cycles exhibit interestingly different behaviour compared to tight Hamilton cycles. A key part of the proof is a precise characterisation of all 4-uniform hypergraphs H on n vertices with delta(H) >= n/2 - epsilon * n which do not contain a Hamilton 2-cycle; this may be of independent interest. As an additional corollary of this characterisation, we obtain an exact Dirac-type bound for the existence of a Hamilton 2-cycle in a large 4-uniform hypergraph.

Motivated by work of Haythorpe, Thomassen and the author showed that there exists a positive constant $c$ such that there is an infinite family of 4-regular 4-connected graphs, each containing exactly $c$ Hamiltonian cycles. We complement this by proving that the same conclusion holds for planar 4-regular 3-connected graphs, although it does not hold for planar 4-regular 4-connected graphs by a result of Brinkmann and Van Cleemput [European J. Combin., 97 (2021), 103395], and that it holds for 4-regular graphs of connectivity 2 with the constant $144 < c$, which we believe to be minimal among all Hamiltonian 4-regular graphs of sufficiently large order. We then disprove a conjecture of Haythorpe by showing that for every nonnegative integer $k$ there is a 5-regular graph on $26 + 6k$ vertices with $2^{k+10} \cdot 3^{k+3}$ Hamiltonian cycles. We prove that for every $d \ge 3$ there is an infinite family of Hamiltonian 3-connected graphs with minimum degree $d$, with a bounded number of Hamiltonian cycles. It is shown that if a 3-regular graph $G$ has a unique longest cycle $C$, at least two components of $G - E(C)$ have an odd number of vertices on $C$, and that there exist 3-regular graphs with exactly two such components.

Consider a 3-uniform hypergraph H with n vertices. A tight Hamilton cycle C ⊂ H is a collection of n edges for which there is an ordering of the vertices v1,..., vn where every triple of consecutive vertices {vi, vi+1, vi+2} is an edge of C (indices considered modulo n). We develop new techniques which show that under certain natural pseudo-random conditions, almost all edges of H can be covered by edge-disjoint tight Hamilton cycles, for n divisible by 4. Consequently, random 3-uniform hypergraphs can be almost completely packed with tight Hamilton cycles whp, for n divisible by 4 and p not too small. Along the way, we develop a similar result for packing Hamilton cycles in pseudo-random digraphs with even numbers of vertices.

The n-dimensional crossed cube CQn, a variation of the hypercube Qn, has the same number of vertices and the same number of edges as Qn, but it has only about half of the diameter of Qn. In the interconnection network, some efficient communication algorithms can be designed based on edge-disjoint Hamiltonian cycles. In addition, two edge-disjoint Hamiltonian cycles also provide the edge-fault tolerant Hamiltonicity for the interconnection network. Hung [Discrete Applied Mathematics 181, 109–122, 2015] designed a recursive algorithm to construct two edge-disjoint Hamiltonian cycles on CQn in O(n2n) time. In this paper, we provide an O(n) time algorithm for each vertex in CQn to determine which two edges were used in Hamiltonian cycles 1 and 2, respectively. With the information of each vertex, we can construct two edge-disjoint Hamiltonian cycles in CQn with n ≥ 4.