Search Results (60)

The dimension-balanced cycle (DBC) problem is new in graph theory, with applications such as 3D stereogram reconstruction. In a graph whose edges are partitioned into k dimensions, a cycle is dimension-balanced if edge counts across dimensions differ by at most one. When such a cycle is Hamiltonian, it is called a dimension-balanced Hamiltonian cycle (DBH). Since DBHs do not always exist, a relaxed notion—the weakly dimension-balanced Hamiltonian (WDBH) cycle—was considered, allowing a difference of up to three. We prove that WDBH always exists in any 3-dimensional toroidal mesh graph T_m,n,r for all positive integers m, n, and r. Full article

The rapid growth in demand for diverse network application services has driven the continuous development and expansion of data centers. BCubes was proposed by Microsoft Research Asia for designing modular data centers, and it is a multi-layer recursively constructed network with many advantages. This article shows that BCube is the existence of two edge-disjoint Hamiltonian cycles, abbreviated as two EDHCs, which provide two significant benefits in data center operations: (1) parallel data broadcast and (2) edge fault-tolerance in network communications. We present the following results in this paper: (1) By utilizing the network topology characteristics, we first provide construction algorithms for two EDHCs on low-dimensional BCubes. (2) Based on the algorithm and the recursive structure of BCubes, we prove that two EDHCs exist for all BCubes. (3) Considering all-to-all broadcasting using two EDHCs as transmission channels, we evaluate the performance of all-to-all broadcasting through simulations on low-dimensional BCubes. Full article

Hamiltonian cycle problems play a central role in graph theory and have wide-ranging applications in network-on-chip architectures, interconnection networks, and large-scale parallel systems. When a network contains faulty nodes or faulty links, the feasibility of certain paths becomes restricted, making the construction of Hamiltonian cycles substantially more difficult and increasingly important for ensuring reliable communication. A dimension-balanced Hamiltonian cycle is a special type of cycle that maintains an even distribution of edges across multiple dimensions of a network. Its directed counterpart extends this idea to symmetric directed networks by balancing the number of edges used in each positive and negative direction. Such cycles are desirable because they support uniform traffic distribution and reduce communication contention in practical systems. Previous research has examined the existence of directed dimension-balanced Hamiltonian cycles in directed toroidal mesh networks and has shown that some configurations permit directed dimension-balanced Hamiltonian cycles while others do not. Building on this foundation, this paper investigates the fault-tolerant properties of such networks by analyzing whether directed dimension-balanced Hamiltonian cycles still exist when a single vertex (node) or a single edge (link) is faulty. Our results extend the current understanding of Hamiltonian robustness in symmetric directed networks. Full article

Interconnection networks (often modeled as graphs) are critical for high-performance computing systems, as they have significant impact on performance metrics like latency and bandwidth. The dragonfly network, denoted as $D(n,r)$, is a promising topology owing to its modularity, low diameter, and cost-effectiveness. Ensuring reliability and efficiency in these networks requires robust cycle embedding properties. The two-disjoint-cycle-cover pancyclicity ensures that the network can be partitioned into two vertex-disjoint cycles of any feasible length. This suggests potential advantages for improving fault tolerance and load balancing strategies in interconnection networks. Formally, a graph G is called two-disjoint-cycle-cover $[a_1,a_2]$-pancyclic if for any integer ℓ satisfying $a_1\le 𝓁\le a_2$, there exist two vertex-disjoint cycles $C_1$ and $C_2$ in G such that $|V(C_1)|=𝓁$ and $|V(C_2)|=|V(G)|-𝓁$. While prior work has established Hamiltonicity and pancyclicity for $D(n,r)$, the two-disjoint-cycle-cover problem remains unexplored. This paper fills this gap by proving that $D(n,r)$ is two-disjoint-cycle-cover $[3,⌊{\displaystyle \frac{|V(D(n,r))|}{2}}⌋]$-pancyclic with $n\ge 3$ and $r\ge 2$, generalizing existing knowledge. Moreover, it can be obtained that $D(n,r)$ is vertex-disjoint-cycle-coverable. Our proof employs a constructive method with case analysis, ensuring the existence of such cycles. Full article

Conservation Tillage Technology (CTT) is vital for mitigating soil degradation, yet its adoption rates remain far below targets. This study develops an evolutionary game model that integrates heterogeneous time preferences and the lemon market effect to explore the dynamic adoption mechanisms among boundedly rational farmers. Results show that farmers with high discount rates (indicating strong time preference) undervalue long-term benefits, creating a significant barrier to CTT adoption. The lemon market effect, where P represents the benefit from information asymmetry for non-adopters and Q is the corresponding loss for adopters, critically shapes the system equilibria: (1) when $P>Q$, a stable coexistence of adoption strategies emerges; (2) when $P<Q$, the system exhibits unpredictable heteroclinic cycles; (3) when $P=Q$, it forms a conservative Hamiltonian system characterized by stable periodic oscillations. These findings provide a dynamic analytical framework for understanding green technology diffusion and offer a theoretical basis for crafting sustainable agricultural policies in developing countries. Full article

This paper investigates the decomposition of the $\lambda$-fold complete 3-uniform hypergraph $\lambda K_{\nu }^{\left(3\right)}$ into 4-cycles, denoted as $S_{\lambda }(3,{\mathrm{\Gamma }}_{5,1},v)$. Using the ${\mathrm{\Gamma }}_{5,1}$-structure as a model, we develop recursive construction techniques that exploit symmetric properties and provide explicit designs for small orders. These recursive frameworks enable the systematic generation of large-order hypergraph designs from smaller building blocks, illustrating the symmetric inheritance of structural properties. We establish that the necessary conditions for such a decomposition are also sufficient: an $S_{\lambda }(3,{\mathrm{\Gamma }}_{5,1},v)$ exists if and only if $24\mid \lambda v(v-1)(v-2),2\mid \lambda (v-1)(v-2),\mathrm{and}v\ge 5.$ This result highlights the deep interplay between combinatorial design theory and symmetry in hypergraph decompositions. Full article

Aperiodic tessellations of polykite unitiles, such as hats and turtles, and the recently introduced hares, red squirrels, and gray squirrels, have attracted significant interest due to their structural and combinatorial properties. Our primary objective here is to learn how we could build a self-assembling polyhedron that would have an aperiodic tessellation of its surface using only a single type of polykite unitile. Such a structure would be analogous to some viral capsids that have been reported to have a quasicrystal configuration of capsomeres. We report on our use of a graph–theoretic approach to examine the adjacency and symmetry constraints of these unitiles in tessellations because by using graph theory rather than the usual geometric description of polykite unitiles, we are able (1) to identify which particular vertices and/or edges join one another in aperiodic tessellations; (2) to take advantage of being scale invariant; and (3) to use the deformability of shapes in moving from the plane to the sphere. We systematically classify their connectivity patterns and structural characteristics by utilizing Hamiltonian cycles of vertex degrees along the perimeters of the unitiles. In addition, we applied Blumeyer’s 2 × 2 classification framework to investigate the influence of chirality and periodicity, while Heesch numbers of corona structures provide further insights into tiling patterns. Furthermore, we analyzed the distribution of polykite unitiles with Voronoi tessellations and their Delaunay triangulations. The results of this study contribute to a better understanding of self-assembling structures with potential applications in biomimetic materials, nanotechnology, and synthetic biology. Full article

The hypercube $Q_n$ is a well-known and efficient interconnection network. Ruskey and Savage posed the following question: does every matching in a hypercube $Q_n$ for $n\ge 2$ extend to a Hamiltonian cycle? Fink addressed this by proving that every perfect matching extends to a Hamiltonian cycle in $Q_n$, thereby resolving Kreweras’ conjecture. Ruskey and Savage’s problem is still open and has been proven only for small matchings. An edge of $Q_n$ is an i-edge when the binary representations of its endpoints differ at the ith coordinate. In this paper, we consider $Q_n$ for $n\ge 3$ and show that any matching consisting of edges of at most six types, which does not cover every pair of vertices at a distance of 3, extends to a Hamiltonian cycle. Full article

Topological transitions are relevant for boundary conditions. Therefore, we investigate the bulk spectra, edge states, and topological phases of Szegedy’s quantum search on a one-dimensional (1D) cycle with self-loops, where the search operator can be formulated as an open boundary condition. By establishing an equivalence with coined quantum walks (QWs), we analytically derive and numerically illustrate the quasienergies dispersion relations of bulk spectra and edge states for Szegedy’s quantum search. Interestingly, novel gapless three-band structures are observed, featuring a flat band and three-fold degenerate points. We identify the topological phases $\pm 2$ as the Chern number. This invariant is computed by leveraging chiral symmetry in zero diagonal Hermitian Hamiltonians that satisfy our quasienergies constraints. Furthermore, we demonstrate that the edge states enhance searches on the marked vertices, while the nontrivial bulk spectra facilitate ballistic spread for Szegedy’s quantum search. Crucially, we find that gapless topological phases arise from three-fold degenerate points and are protected by chiral symmetry, distinguishing ill-defined topological transition boundaries. Full article

Given a graph G = (V, E), the edge set can be partitioned into k dimensions, for a positive integer k. The set of all i-dimensional edges of G is a subset of E(G) denoted by E_i. A Hamiltonian cycle C on G contains all vertices on G. Let E_i(C) = E(C) ∩ E_i. For any 1 ≤ i ≤ k, C is called a dimension-balanced Hamiltonian cycle (DBH, for short) on G if ||E_i(C)| − |E_j(C)|| ≤ 1 for all 1 ≤ i < j ≤ k. The dimension-balanced cycle problem is generated with the 3-D scanning problem. Graph G is called p-node q-edge dimension-balanced Hamiltonian (p-node q-edge DBH) if it has a DBH after removing any p nodes and any q edges. G is called h-fault dimension-balanced Hamiltonian (h-fault DBH, for short) if it remains Hamiltonian after removing any h node and/or edges. The design for the network-on-chip (NoC) problem is important. One of the most famous NoC is the toroidal mesh graph T_m,n. The DBC problem on toroidal mesh graph T_m,n is appropriate for designing simple algorithms with low communication costs and avoiding congestion. Recently, the problem of a one-fault DBH on T_m,n has been studied. This paper solves the one-node one-edge DBH problem in the two-fault DBH problem on T_m,n. Full article

Finding a Hamiltonian cycle in a graph G = (V, E) is a well-known problem. The challenge of finding a Hamiltonian cycle that avoids these faults when faulty vertices or edges are present has been extensively studied. When the edge set of G is partitioned into k dimensions, the problem of dimension-balanced Hamiltonian cycles arises, where the Hamiltonian cycle uses approximately the same number of edges from each dimension (differing by at most one). This paper studies whether a dimension-balanced Hamiltonian cycle (DBH) exists in toroidal mesh graphs T_m,n when a single vertex or edge is faulty, called the one-fault DBH problem. We establish that T_m,n is one-fault DBH, except in the following cases: (1) both m and n are even; (2) one of m and n is 3, while the other satisfies mod 4 = 3 and is greater than 6; (3) one of m and n is odd, while the other satisfies mod 4 = 2. Additionally, this paper resolves a conjecture from prior literature, thereby providing a complete solution to the DBP problem on T_m,n. Full article

Let $G=(V,E)$ be a graph. The first Zagreb index of a graph G is defined as ${\sum }_{u\in V}{d}_G^2\left(u\right)$, where $d_G\left(u\right)$ is the degree of vertex u in G. A graph G is called Hamiltonian (resp. traceable) if G has a cycle (resp. path) containing all the vertices of G. Using two established inequalities, in this paper, we present sufficient conditions involving the first Zagreb index for Hamiltonian graphs and traceable graphs. We also present upper bounds for the first Zagreb index of a graph and characterize the graphs achieving the upper bounds. Full article

This research is focused on evolutionary algorithms, with genetic and memetic algorithms discussed in more detail. A graph theory problem related to finding a minimal Hamiltonian cycle in a complete undirected graph (Travelling Salesman Problem—TSP) is considered. The implementations of two approximate algorithms for solving this problem, genetic and memetic, are presented. The main objective of this study is to determine the influence of the local search method versus the influence of the genetic crossover operator on the quality of the solutions generated by the memetic algorithm for the same input data. The results show that when the number of possible Hamiltonian cycles in a graph is increased, the memetic algorithm finds better solutions. The execution time of both algorithms is comparable. Also, the number of solutions that mutated during the execution of the genetic algorithm exceeds 50% of the total number of all solutions generated by the crossover operator. In the memetic algorithm, the number of solutions that mutate does not exceed 10% of the total number of all solutions generated by the crossover operator, summed with those of the local search method. Full article

In this paper, two classes of near-Hamiltonian systems with a nilpotent center are considered: the coexistence of algebraic limit cycles and small limit cycles. For the first class of systems, there exist $2n+1$ limit cycles, which include an algebraic limit cycle and $2n$ small limit cycles. For the second class of systems, there exist $\frac{n^2+3n+2}{2}$ limit cycles, including an algebraic limit cycle and $\frac{n^2+3n}{2}$ small limit cycles. Full article

Embedding cycles into a network topology is crucial for a network simulation. In particular, embedding Hamiltonian cycles is a major requirement for designing good interconnection networks. A graph G is called r-spanning cyclable if, for any r distinct vertices $v_1,v_2,\dots ,v_r$ of G, there exist r cycles $C_1,C_2,\dots ,C_r$ in G such that $v_i$ is on $C_i$ for every i, and every vertex of G is on exactly one cycle $C_i$. If $r=1$, this is the classical Hamiltonian problem. In this paper, we focus on the problem of embedding spanning disjoint cycles in bipartite k-ary n-cubes. Let $k\ge 4$ be even and $n\ge 2$. It is shown that the n-dimensional bipartite k-ary n-cube $Q_n^k$ is m-spanning cyclable with $m\le 2n-1$. Considering the degree of $Q_n^k$, the result is optimal. Full article