Search Results (238)

With the proliferation of smart devices and the emergence of high-bandwidth applications, Unmanned Aerial Vehicle (UAV)-assisted Device-to-Device (D2D) communications and Non-Orthogonal Multiple Access (NOMA) technologies are increasingly becoming important means of coping with the scarcity of the spectrum and with high data demand in future wireless networks. However, the efficient coordination of these techniques in complex and changing 3D environments still faces many challenges. To this end, this paper proposes a NOMA-based multi-UAV-assisted D2D communication model in which multiple UAVs are deployed in 3D space to act as airborne base stations to serve ground-based cellular users with D2D clusters. In order to maximize the system throughput, this study constructs an optimization problem of joint channel assignment, trajectory design, and power control, and on the basis of these points, this study proposes a joint dynamic hypergraph Multi-Agent Deep Q Network (DH-MDQN) algorithm. The dynamic hypergraph method is first used to construct dynamic simple edges and hyperedges and to transform them into directed graphs for efficient dynamic coloring to optimize the channel allocation process; subsequently, in terms of trajectory design and power control, the problem is modeled as a multi-agent Markov Decision Process (MDP), and the Multi-Agent Deep Q Network (MDQN) algorithm is used to collaboratively determine the trajectory design and power control of the UAVs. Simulation results show the following: (1) the proposed algorithm can achieve higher system throughput than several other benchmark algorithms with different numbers of D2D clusters, different D2D cluster communication spacing, and different UAV sizes; (2) the proposed algorithm designs UAV trajectory optimization with a 27% improvement in system throughput compared to the 2D trajectory; and (3) in the NOMA scenario, compared to the case of no decoding order constraints, the system throughput shows on average a 34% improvement. Full article

This paper investigates the application of graph theory and variants of greedy graph coloring algorithms for the optimization of distributed peer-to-peer networks, with a special focus on private blockchain networks. The graph coloring problem, as an NP-hard problem, presents a challenge in determining the minimum number of colors needed to efficiently allocate resources within the network. The paper deals with the influence of different graph density, i.e., the number of links, on the efficiency of greedy algorithms such as DSATUR, Descending, and Ascending. Experimental results show that increasing the number of links in the network contributes to a more uniform distribution of colors and increases the resistance of the network, whereby the DSATUR algorithm achieves the most uniform color saturation. The optimal configuration for a 100-node network has been identified at around 2000 to 2500 links, which achieves stability without excessive redundancy. These results are applied in the context of a private blockchain network that uses optimal connectivity to achieve high resilience and efficient resource allocation. The research findings suggest that adapting network configuration using greedy algorithms can contribute to the optimization of distributed systems, making them more stable and resilient to loads. Full article

An edge-coloring σ of a connected graph G is called rainbow if there exists a rainbow path connecting any pair of vertices. In contrast, σ is monochromatic if there is a monochromatic path between any two vertices. Some graphs can admit a coloring which is simultaneously rainbow and monochromatic; for instance, any coloring of Kn is rainbow and monochromatic. This paper refers to such a coloring as dual coloring. We investigate dual coloring on various graphs and raise some questions about the sufficient conditions for connected graphs to be dual connected. Full article

In order to resolve Borodin’s Conjecture, DP-coloring was introduced in 2017 to extend the concept of list coloring. In previous works, it is proved that every planar graph without 7-cycles and butterflies is DP-4-colorable. And any planar graph that does not have 5-cycle adjacent to 6-cycle is DP-4-colorable. The existing research mainly focus on the forbidden adjacent cycles that guarantee the DP-4-colorability for planar graph. In this paper, we demonstrate that any planar graph G that excludes 7-cycles adjacent to k-cycles (for each k=4,5), and does not feature a Near-bow-tie as an induced subgraph, is DP-4-colorable. This result extends the findings of the previous works mentioned above. Full article

In this paper, we explore the connection between sensor networks and graph theory. Sensor networks represent distributed systems of interconnected devices that collect and transmit data, while graph theory provides a robust framework for modeling and analyzing complex networks. Specifically, we focus on vertex coloring, Eulerian paths, and Hamiltonian paths within the Delaunay graph associated with a sensor network. These concepts have critical applications in sensor networks, including connectivity analysis, efficient data collection, route optimization, task scheduling, and resource management. We derive theoretical results related to the chromatic number and the existence of Eulerian and Hamiltonian trails in the graph linked to the sensor network. Additionally, we complement this theoretical study with the implementation of several algorithmic procedures. A case study involving the monitoring of a sugarcane field, coupled with a computational analysis, demonstrates the performance and practical applicability of these algorithms in real-world scenarios. Full article

Let G=(V,E) be a connected graph. A bijection f:E→{1,…,|E|} is called a local antimagic labeling if, for any two adjacent vertices x and y, f+(x)≠f+(y), where f+(x)=∑e∈E(x)f(e), and E(x) is the set of edges incident to x. Thus, a local antimagic labeling induces a proper vertex coloring of G, where the vertex x is assigned the color f+(x). The local antimagic chromatic number χla(G) is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper, we present some families of bridge graphs with χla(G)=3 and give several ways to construct bridge graphs with χla(G)=3. Full article

Ramsey’s theorem states that for any natural numbers n, m there exists a natural number N such that any red–blue coloring of the graph KN contains either a red Kn or blue Km as a subgraph. The smallest such N is called the Ramsey number, denoted as R(n,m). In this paper, we reformulate this theorem and present a proof of Ramsey’s theorem that is novel as far as we are aware. Full article

Practical algorithms for computing canonical forms of multi-digraphs do not exist in the literature. This paper proposes two practical approaches for finding canonical forms, from the perspective of nD symbolic computation. Initially, the approaches turn the problem of finding canonical forms of multi-digraphs into computing canonical forms of indexed monomials in computer algebra. Then, the first approach utilizes the double coset representative method in computational group theory for canonicalization of indexed monomials and shows that finding the canonical forms of a class of multi-digraphs in practice has polynomial complexity of approximately O((k+p)2) or O(k2.1) by the computer algebra system (CAS) tool Tensor-canonicalizer. The second approach verifies the equivalence of canonicalization of indexed monomials and finding canonical forms of (simple) colored tripartite graphs. It is found that the proposed algorithm takes approximately O((k+2p)4.803) time for a class of multi-digraphs in practical implementation, combined with one of the best known graph isomorphism tools Traces, where k and p are the vertex number and edge number of a multi-digraph, respectively. Full article

An even ring system G is a simple 2-connected plane graph with all interior vertices of degree 3, all exterior vertices of either degree 2 or 3, and all finite faces of an even length. G is angularly connected if all of the peripheral segments of G have odd lengths. In this paper, we show that every angularly connected even ring system G, which does not contain any triple of altogether-adjacent peripheral faces, has a perfect matching. This was achieved by finding an appropriate edge coloring of G, derived from the proof of the existence of a proper face 3-coloring of the graph. Additionally, an infinite family of graphs that are face 3-colorable has been identified. When interpreted in the context of the inner dual of G, this leads to the introduction of 3-colorable graphs containing cycles of lengths 4 and 6, which is a supplementation of some already known results. Finally, we have investigated the concept of the Clar structure and Clar set within the aforementioned family of graphs. We found that a Clar set of an angularly connected even ring system cannot in general be obtained by minimizing the cardinality of the set A. This result is in contrast to the previously known case for the subfamily of benzenoid systems, which admit a face 3-coloring. Our results open up avenues for further research into the properties of Clar and Fries sets of angularly connected even ring systems. Full article

2D-3D registration is increasingly being applied in various scientific and engineering scenarios. However, due to appearance differences and cross-modal discrepancies, it is demanding for image and point cloud registration methods to establish correspondences, making 2D-3D registration highly challenging. To handle these problems, we propose a novel and automatic solution for 2D-3D registration in Manhattan world based on line primitives, which we denote as VPPnL. Firstly, we derive the rotation matrix candidates by establishing the vanishing point coordinate system as the link of point cloud principal directions to camera coordinate system. Subsequently, the RANSAC algorithm, which accounts for the clustering of parallel lines, is employed in conjunction with the least-squares method for translation vectors estimation and optimization. Finally, a nonlinear least-squares graph optimization method is carried out to optimize the camera pose and realize the 2D-3D registration and point colorization. Experiments on synthetic data and real-world data illustrate that our proposed algorithm can address the problem of 2D-3D direct registration in the case of Manhattan scenes where images are limited and sparse. Full article

We open here many new tracks of research in anti-Ramsey Theory, considering edge-coloring problems inspired by rainbow coloring and further by odd colorings and conflict-free colorings. Let G be a graph and F any given family of graphs. For every integer n≥|G|, let f(n,G|F) denote the smallest integer k such that any edge coloring of Kn with at least k colors forces a copy of G in which each color class induces a member of F. Observe that in anti-Ramsey problems, each color class is a single edge, i.e., F={K2}. Among the many results introduced in this paper, we mention the following. (1) For every graph G, there exists a constant c=c(G) such that in any edge coloring of Kn with at least cn colors there is a copy of G in which every vertex v is incident with an edge whose color appears only once among all edges incident with v. (2) In sharp contrast to the above result we prove that if F is the class of all odd graphs (having vertices with odd degrees only) then f(n,Kk|F)=(1+o(1))ex(n,K⌈k/2⌉), which is quadratic for k≥5. (3) We exactly determine f(n,G|F) for small graphs when F belongs to several families representing various odd/even coloring constraints. Full article

We propose a Ramsey-theory-based approach for the analysis of the behavior of isolated mechanical systems containing interacting particles. The total momentum of the system in the frame of the center of masses is zero. The mechanical system is described by a Ramsey-theory-based, bi-colored, complete graph. Vectors of momenta of the particles p→i serve as the vertices of the graph. We start from the graph representing the system in the frame of the center of masses, where the momenta of the particles in this system are p→cmi. If (p→cmi(t)·p→cmj(t))≥0 is true, the vectors of momenta of the particles numbered i and j are connected with a red link; if (p→cmi(t)·p→cmj(t))<0 takes place, the vectors of momenta are connected with a green link. Thus, the complete, bi-colored graph emerges. Considering an isolated system built of six interacting particles, according to the Ramsey theorem, the graph inevitably comprises at least one monochromatic triangle. The coloring procedure is invariant relative to the rotations/translations of frames; thus, the graph representing the system contains at least one monochromatic triangle in any of the frames emerging from the rotation/translation of the original frame. This gives rise to a novel kind of mechanical invariant. Similar coloring is introduced for the angular momenta of the particles. However, the coloring procedure is sensitive to Galilean/Lorenz transformations. Extensions of the suggested approach are discussed. Full article

We applied the Ramsey analysis to the sets of points belonging to Riemannian manifolds. The points are connected with two kinds of lines: geodesic and non-geodesic. This interconnection between the points is mapped into the bi-colored, complete Ramsey graph. The selected points correspond to the vertices of the graph, which are connected with the bi-colored links. The complete bi-colored graph containing six vertices inevitably contains at least one mono-colored triangle; hence, a mono-colored triangle, built of the green or red links, i.e., non-geodesic or geodesic lines, consequently appears in the graph. We also considered the bi-colored, complete Ramsey graphs emerging from the intersection of two Riemannian manifolds. Two Riemannian manifolds, namely (M1,g1) and (M2,g2), represented by the Riemann surfaces which intersect along the curve (M1,g1)∩(M2,g2)=ℒ were addressed. Curve ℒ does not contain geodesic lines in either of the manifolds(M1,g1) and (M2,g2). Consider six points located on the ℒ: {1,…6}⊂ℒ. The points {1,…6}⊂ℒ are connected with two distinguishable kinds of the geodesic lines, namely with the geodesic lines belonging to the Riemannian manifold (M1,g1)/red links, and, alternatively, with the geodesic lines belonging to the manifold (M2,g2)/green links. Points {1,…6}⊂ℒ form the vertices of the complete graph, connected with two kinds of links. The emerging graph contains at least one closed geodesic line. The extension of the theorem to the Riemann surfaces of various Euler characteristics is presented. Full article

The security of traditional public key cryptography algorithms depends on the difficulty of the underlying mathematical problems. Asymmetric topological encryption is a graph-dependent encryption algorithm produced to resist attacks by quantum computers on these mathematical problems. The security of this encryption algorithm depends on two types of NP-complete problems: subgraph isomorphism and graph coloring. Topological coding technology refers to the technology of generating key strings or topology signature strings through topological coding graphs. We take odd-graceful labeling and set-ordered odd-graceful labeling as limiting functions, and propose two kinds of topological coding generation technique, which we call the random leaf-adding operation and randomly adding edge-removing operation. Through these two techniques, graphs of the same scale and larger scales can be generated with the same type of labeling so as to derive more number strings, expand the key space, and analyze the topology and property of the generated graphs. Full article

Quantum annealers conventionally use forward annealing to generate heuristic solutions. Reverse annealing can potentially generate better solutions but necessitates an appropriate initial state. Ways to find such states are generally unknown or highly problem dependent, offer limited success, and severely restrict the scope of reverse annealing. We use a general method that improves the overall solution quality and quantity by feeding reverse annealing with low-quality solutions obtained from forward annealing. An experimental demonstration of solving the graph coloring problem using the D-Wave quantum annealers shows that our method is able to convert invalid solutions obtained from forward annealing to at least one valid solution obtained after assisted reverse annealing for 57% of 459 random Erdos–Rényi graphs. Our method significantly outperforms random initial states, obtains more unique solutions on average, and widens the applicability of reverse annealing. Although the average number of valid solutions obtained drops exponentially with the problem size, a scaling analysis for the graph coloring problem shows that our method effectively extends the computational reach of conventional forward annealing using reverse annealing. Full article