Hamiltonian Cycle Research Articles

A 3/2-approximation algorithm for the multiple Hamiltonian path problem with no prefixed endpoints

We study the multiple Hamiltonian path problem (MHPP) defined on a complete undirected graph G with n vertices. The edge weights of G are non-negative and satisfy the triangle inequality. The MHPP seeks to find a collection of k paths with exactly one visit to each vertex of G with the minimum total edge weight, where endpoints of the paths are not prefixed. We present a 3/2-approximation algorithm for MHPP with time complexity O(n3) for arbitrary k≥1.

On sufficient conditions for spanning structures in dense graphs

AbstractWe study structural conditions in dense graphs that guarantee the existence of vertex‐spanning substructures such as Hamilton cycles. It is easy to see that every Hamiltonian graph is connected, has a perfect fractional matching and, excluding the bipartite case, contains an odd cycle. A simple consequence of the Robust Expander Theorem of Kühn, Osthus and Treglown tells us that any large enough graph that robustly satisfies these properties must already be Hamiltonian. Our main result generalises this phenomenon to powers of cycles and graphs of sublinear bandwidth subject to natural generalisations of connectivity, matchings and odd cycles. This answers a question of Ebsen, Maesaka, Reiher, Schacht and Schülke and solves the embedding problem that underlies multiple lines of research on sufficient conditions for spanning structures in dense graphs. As applications, we recover and establish Bandwidth Theorems in a variety of settings including Ore‐type degree conditions, Pósa‐type degree conditions, deficiency‐type conditions, locally dense and inseparable graphs, multipartite graphs as well as robust expanders.

Two completely independent spanning trees of split graphs

G is a split graph if V(G) can be partitioned into a clique D and an independent set I. Let τ(G) denote the toughness of a graph G. If a graph G contains two spanning trees T1,T2 such that for each two distinct vertices x,y of G, the (x,y)-path in each Ti has no common edge and no common vertex except for the two ends, then T1,T2 are called two completely independent spanning trees (CISTs) of G,i∈{1,2}. Several results have shown that some sufficient conditions for Hamiltonian graphs also guarantee the existence of two CISTs, which implies the Hamiltonicity and the existence of two CISTs of a graph have close connection. In this paper, we prove that if G is a Hamiltonian split graph with |D|>max{3,|I|}, then G contains two CISTs. Moreover, we show that if G is a Hamiltonian split graph with τ(G)>1, then G contains two CISTs. As a corollary, we obtain that every 32-tough split graph contains two CISTs.

Cycles of quadratic Latin squares and antiperfect 1‐factorisations

AbstractA Latin square of order is an matrix of symbols, such that each symbol occurs exactly once in each row and column. For an odd prime power let denote the finite field of order . A quadratic Latin square is a Latin square defined by for some such that and are quadratic residues in . Quadratic Latin squares have previously been used to construct perfect 1‐factorisations, mutually orthogonal Latin squares and atomic Latin squares. We first characterise quadratic Latin squares which are devoid of Latin subsquares. Let be a graph and a 1‐factorisation of . If the union of every pair of 1‐factors in induces a Hamiltonian cycle in then is called perfect, and if there is no pair of 1‐factors in which induce a Hamiltonian cycle in then is called antiperfect. We use quadratic Latin squares to construct new examples of antiperfect 1‐factorisations of complete graphs and complete bipartite graphs. We also demonstrate that for each odd prime , there are only finitely many orders , which are powers of , such that quadratic Latin squares of order could be used to construct perfect 1‐factorisations of complete graphs or complete bipartite graphs.

Powers of Hamiltonian cycles in randomly augmented Dirac graphs—The complete collection

AbstractWe study the powers of Hamiltonian cycles in randomly augmented Dirac graphs, that is, ‐vertex graphs with minimum degree at least to which some random edges are added. For any Dirac graph and every integer , we accurately estimate the threshold probability for the event that the random augmentation contains the ‐th power of a Hamiltonian cycle.

New Characteristics of AI Analysis of Financial Risks and their Transmission Traceability Paths

New Characteristics of AI Analysis of Financial Risks and their Transmission Traceability Paths

Hamiltonian path, routing, broadcasting algorithms for connected square network graphs

Connected Square Network Graphs (CSNG) in the study of Selcuk (2022) and Selcuk and Tankul (2022) is reconsidered in this paper. Although (CSNG) is a 2-dimensional mesh structure, the most important feature of this graph is that it is a hypercube variant. For this reason, this study focuses on development algorithms that find solutions to various problems for (CSNG) with the help of hypercube. Firstly, an efficient algorithm that finds the Hamiltonian path is given. Further, two different algorithms that perform the mapping of labels in graph and the unicast routing are given. Furthermore, the parallel process for mapping and unicast routing is discussed. Finally, guidelines are given for broadcasting algorithms.

Hamiltonian paths and Hamiltonian cycles passing through prescribed linear forests in star graph with fault-tolerant edges

Hamiltonian paths and Hamiltonian cycles passing through prescribed linear forests in star graph with fault-tolerant edges

AlphaSnake: Policy Iteration on a Nondeterministic NP-Hard Markov Decision Process (Student Abstract)

Reinforcement learning has been used to approach well-known NP-hard combinatorial problems in graph theory. Among these, Hamiltonian cycle problems are exceptionally difficult to analyze, even when restricted to individual instances of structurally complex graphs. In this paper, we use Monte Carlo Tree Search (MCTS), the search algorithm behind many state-of-the-art reinforcement learning algorithms such as AlphaZero, to create autonomous agents that learn to play the game of Snake, a game centered on properties of Hamiltonian cycles on grid graphs. The game of Snake can be formulated as a single-player discounted Markov Decision Process (MDP), where the agent must behave optimally in a stochastic environment. Determining the optimal policy for Snake, defined as the policy that maximizes the probability of winning -- or win rate -- with higher priority and minimizes the expected number of time steps to win with lower priority, is conjectured to be NP-hard. Performance-wise, compared to prior work in the Snake game, our algorithm is the first to achieve a win rate over 0.5 (a uniform random policy achieves a win rate < 2.57 x 10^{-15}), demonstrating the versatility of AlphaZero in tackling NP-hard problems.

Recursive definition and two edge-disjoint Hamiltonian cycles of bubble-sort star graphs

The bubble-sort star graph is an interconnection network for multiprocessor systems. Recursive structure is important for interconnection networks, since it can reduce the complex cases into simple cases, and it can keep good properties independent of dimensions. Many algorithms use the idea of recursive construction. Edge-disjoint Hamiltonian cycles not only provide the basis of all-to-all broadcasting algorithm for networks but also provide a replaceable Hamiltonian cycle for transmission when the other Hamiltonian cycle contains faulty edges in an interconnection network. In this paper, we propose the recursive definition of bubble-sort star graphs. Then as an application, we obtain two edge-disjoint Hamiltonian cycles of bubble-sort star graphs.

Antidirected Hamiltonian Paths and Cycles of Digraphs with $$\alpha _{2}$$-Stable Number 2

Antidirected Hamiltonian Paths and Cycles of Digraphs with $$\alpha _{2}$$-Stable Number 2

Cycle lengths in randomly perturbed graphs

AbstractLet be an ‐vertex graph, where for some . A result of Bohman, Frieze and Martin from 2003 asserts that if , then perturbing via the addition of random edges, a.a.s. yields a Hamiltonian graph. We prove several improvements and extensions of the aforementioned result. In particular, keeping the bound on as above and allowing for , we determine the correct order of magnitude of the number of random edges whose addition to a.a.s. yields a pancyclic graph. Moreover, we prove similar results for sparser graphs, and assuming the correctness of Chvátal's toughness conjecture, we handle graphs having larger independent sets. Finally, under milder conditions, we determine the correct order of magnitude of the number of random edges whose addition to a.a.s. yields a graph containing an almost spanning cycle.

Counting Traversing Hamiltonian Cycles in Tiled Graphs

Recently, the problem of counting Hamiltonian cycles in 2-tiled graphs was resolved by Vegi Kalamar, Bokal, and Žerak. In this paper, we continue our research on generalized tiled graphs. We extend algorithms on counting traversing Hamiltonian cycles from 2-tiled graphs to generalized tiled graphs. We further show that, similarly as for 2-tiled graphs, for a fixed finite set of tiles, counting traversing Hamiltonian cycles can be performed in linear time with respect to the size of such graph, implying that counting traversing Hamiltonian cycles in tiled graphs is fixed-parameter tractable.

Spanning -cycles in random graphs

AbstractWe extend a recent argument of Kahn, Narayanan and Park ((2021) Proceedings of the AMS 149 3201–3208) about the threshold for the appearance of the square of a Hamilton cycle to other spanning structures. In particular, for any spanning graph, we give a sufficient condition under which we may determine its threshold. As an application, we find the threshold for a set of cyclically ordered copies of $C_4$ that span the entire vertex set, so that any two consecutive copies overlap in exactly one edge and all overlapping edges are disjoint. This answers a question of Frieze. We also determine the threshold for edge-overlapping spanning $K_r$ -cycles.

A smoother notion of spread hypergraphs

AbstractAlweiss, Lovett, Wu, and Zhang introduced $q$ -spread hypergraphs in their breakthrough work regarding the sunflower conjecture, and since then $q$ -spread hypergraphs have been used to give short proofs of several outstanding problems in probabilistic combinatorics. A variant of $q$ -spread hypergraphs was implicitly used by Kahn, Narayanan, and Park to determine the threshold for when a square of a Hamiltonian cycle appears in the random graph $G_{n,p}$ . In this paper, we give a common generalization of the original notion of $q$ -spread hypergraphs and the variant used by Kahn, Narayanan, and Park.

Tangled Cord of FTTM4

Fuzzy Topological Topographic Mapping (FTTM) is a mathematical model that consists of a set of homeomorphic topological spaces designed to solve the neuro magnetic inverse problem. A sequence of FTTM, denoted as FTTMn, is an extension of FTTM that is arranged in a symmetrical form. The special characteristic of FTTM, namely the homeomorphisms between its components, allows the generation of new FTTM. Later, the FTTMn can also be viewed as a graph. Previously, a group of researchers defined an assembly graph and utilized it to model a DNA recombination process. Some researchers then used this to introduce the concept of tangled cords for assembly graphs. In this paper, the tangled cord for FTTM4 is used to calculate the Eulerian paths. Furthermore, it is utilized to determine the least upper bound of the Hamiltonian paths of its assembly graph. Hence, this study verifies the conjecture made by Burns et al.

The speed and threshold of the biased perfect matching and Hamilton cycle games

The speed and threshold of the biased perfect matching and Hamilton cycle games

Reconfiguration of Hamiltonian Cycles in Rectangular Grid Graphs

We study reconfiguration of simple Hamiltonian cycles in a rectangular grid graph [Formula: see text], where the Hamiltonian cycle in each step of the reconfiguration connects every internal vertex of [Formula: see text] to a boundary vertex by a single straight line segment. We introduce two operations, flip and transpose, which are local to the grid. We show that any simple cycle of [Formula: see text] can be reconfigured to any other simple cycle of [Formula: see text] using [Formula: see text] flip and transpose operations. Our result proves that the simple Hamiltonian cycle graph [Formula: see text] is connected with respect to those two operations and has diameter [Formula: see text].

A Hamiltonian graph model for the cooperative toughening of crystalline phases and covalent adaptable networks in semi-crystalline thermoset epoxy

The existence of bond exchange reactions and covalent adaptable networks (CANs) in thermoset epoxy has facilitated its self-healing and reversible mechanical capabilities. However, the toughening mechanisms and cooperative coupling of these crystal phases and CANs in a semi-crystalline thermoset epoxy have not been well understood. In this study, a Hamiltonian graph model is formulated to examine toughening mechanisms in the semi-crystalline thermoset epoxy based on the vertices and paths, both of which are employed to describe the crystalline phases and CANs, respectively. A free-energy equation is also developed based on the tail and tie free energy functions to investigate the cooperative coupling of crystal phases and CANs. The crystal phases increase the cross-linking density of the CANs, which helps the crystal phases with a homogeneous dispersion. Moreover, an extended Maxwell model is developed along with the Hamiltonian graph to explore the coupling effect of crystal phase and CAN on the mechanical behaviors of semi-crystalline thermoset epoxy. A constitutive stress–strain relationship is then proposed to describe the self-healing and toughening behaviors of semi-crystalline thermoset epoxy. The stress–strain relationship of semi-crystalline polymers, which incorporates crystal phases and CANs, has been thoroughly investigated using the analytical results obtained from the proposed Hamiltonian graph model. Finally, the effectiveness of the proposed model is verified using the finite element analysis method and a set of experimental data.

Counting Hamilton Cycles in Dirac Hypergraphs

For 0le ell <k, a Hamilton ell -cycle in a k-uniform hypergraph H is a cyclic ordering of the vertices of H in which the edges are segments of length k and every two consecutive edges overlap in exactly ell vertices. We show that for all 0le ell <k-1, every k-graph with minimum co-degree delta n with delta >1/2 has (asymptotically and up to a subexponential factor) at least as many Hamilton ell -cycles as a typical random k-graph with edge-probability delta . This significantly improves a recent result of Glock, Gould, Joos, Kühn and Osthus, and verifies a conjecture of Ferber, Krivelevich and Sudakov for all values 0le ell <k-1.