Eulerian Graphs Research Articles

GRAPHICAL PROPERTIES OF CLUSTERED GRAPHS

Clustering is a strategy for discovering homogeneous clusters in heterogeneous data sets based on comparable structures or properties. The number of nodes or links that must fail for a network to be divided into two or more sub-networks is known as connectivity. In addition to being a metric of network dependability, connectivity also serves as an indicator of performance. The Euler graph can represent almost any issue involving a discrete arrangement of objects. It can be analyzed using the recent field of mathematics called graph theory. This paper discusses the properties of clustered networks like connectivity and chromaticity. Further, the structure of the antipodal graph in the clustered network has been explored.

Vertex Coloring and Eulerian and Hamiltonian Paths of Delaunay Graphs Associated with Sensor Networks

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.

Continuous fiber printing path generation based on Euler diagram division

Continuous fiber printed structures are widely used in aerospace applications because of their excellent performance and high design freedom. Existing path planning methods optimize the continuous fiber printing direction by stress lines or discrete angles, but these methods are far from perfect printing paths and lack planning for connectivity relations. In this study, a planning method for continuous fiber printing paths based on constructing Eulerian circuits from a weighted undirected graph is proposed. The principal stress lines are transformed into a weighted undirected graph, which divides the Eulerian subgraph by splitting the common nodes to plan the continuous fiber printing path circuit. Subsequently, the printing structure parameters are optimized based on simulation analysis. The experimental results, taking MBB (Messerschmitt-Bolkow-Blohm) beams as an example, show that the printing path designed in this paper improves the structural strength by 52.9% and the structural stiffness by 81.7% compared with the conventional printing path.

A new path planning strategy driven by geometric features and tensile properties for 3D printing of continuous fiber reinforced thermoplastic composites

Three-dimensional (3D) printing technology for continuous fiber reinforced thermoplastic composites (C-FRTP), capable of rapid manufacturing of lightweight components with intricate geometric features, has emerged as one of the most promising technologies in the field of advanced composite manufacturing. Path planning is a crucial step for determining the fabrication quality of C-FRTP components. This study proposed a new 3D printing path planning strategy driven by the geometric features and tensile properties of C-FRTP components. The strategy employed the properties of the Euler graph to generate the continuous full-field filling paths, ensuring the geometric features of the target components. The intersections were scattered along the printing path to enhance the tensile strength. The feasibility and advantages of the new path planning strategy were validated by comparative experiments with different printing paths. The results indicated that the new strategy not only achieved the geometric features of the target components but significantly enhanced their tensile strength. Using the printing path generated by the new path planning strategy, the tensile strength of specimens featuring mounting holes reached 349.4 MPa, which was only about 4.1 % lower than the tensile strength of continuous fibers at straight paths. Compared to the existing contour-parallel path, the new strategy in this work improved the tensile properties by about 40.9 %. The new path planning strategy proposed in this study shows great potential to design and fabricate C-FRTP components with enhanced mechanical properties for practical applications.

Optical Image Processing using Eulerian Amplification

Recent advancements in digital photography, particularly through mobile phone cameras, microscopes, and satellite imaging, have made high-resolution image capture increasingly accessible. However, these captured images often contain subtle changes in color resolution and density that are challenging to observe with the naked eye. This study explores the application of Eulerian amplification as a method to detect and enhance such small-amplitude variations, making them visible for improved analysis. Unlike traditional approaches, Eulerian methods are particularly effective for processing smooth structures and high-quality printed materials where minor amplifications in color can reveal significant grading details. Using a pixel-by-pixel Eulerian path analysis, this method allows for a detailed comparison of color intensity values on a fine 8-bit scale, which ranges from 0 to 255. Through this analysis, color distribution patterns are visualized across the spatial structure of the image, which is particularly beneficial for identifying specific variations in quality in printed materials, such as highly graded cards. To manage large image structures, spatial decomposition using a multi-level Gaussian pyramid technique is employed. By applying temporal filtering within select frequency bands (0.4–4 Hz) at a coarse pyramid level, the algorithm effectively pools spatial data, enabling the detection of color density shifts that may indicate image defects. The amplified image data is further processed using a classifier neural network, which provides high sensitivity and specificity in detecting edge defects, image centering issues, scuffs, and breakages. The system demonstrates promising results in identifying true positives while maintaining low false-negative rates, thus confirming the reliability of deep machine learning algorithms in high-sensitivity defect detection. This paper presents the significance of Eulerian amplification in optical image processing, offering a robust tool for precise, automated defect detection across a variety of high-resolution image types.

Completely degenerate equilibria of the Kuramoto model on networks

Kuramoto Networks contain non-hyperbolic equilibria whose stability is sometimes difficult to determine. We consider the extreme case in which all Jacobian eigenvalues are zero. In this case linearizing the system at the equilibrium leads to a Jacobian matrix which is zero in every entry. We call these equilibria completely degenerate. We prove that they exist for certain intrinsic frequencies if and only if the underlying graph is bipartite, and that they do not exist for generic intrinsic frequencies. In the case of zero intrinsic frequencies, we prove that they exist if and only if the graph has an Euler circuit such that the number of steps between any two visits at the same vertex is a multiple of 4. The simplest example is the cycle graph with 4 vertices. We prove that graphs with this property exist for every number of vertices N⩾6 and that they become asymptotically rare for N large. Regarding stability, we prove that for any choice of intrinsic frequencies, any coupling strength and any graph with at least one edge, completely degenerate equilibria are not Lyapunov stable. As a corollary, we obtain that stable equilibria in Kuramoto Networks must have at least one strictly negative eigenvalue.

Critical-edge based tabu search algorithm for solving large-scale multi-vehicle Chinese postman problem

The min–max multi-vehicle Chinese postman problem is an NP-hard problem, which is widely used in path planning problems based on road network graphs, such as urban road structure probing planning, urban road underground cavity detection planning, high-voltage line inspection planning, and so on. With the rapid increase in the number of nodes and connections of road network graph, the solution time and path equilibrium constraints pose new challenges to the problem solving. In this paper, we propose a critical-edge tabu search algorithm, CTA-kroutes, for solving the min–max multi-vehicle postman problem for large-scale road networks. First, the initial solution with balanced path lengths is obtained by segmenting the Eulerian paths; second, the critical edges are moved in the initial solution to construct the neighborhood solution, and the tabu search algorithm is used to find the optimal solution iteratively; and lastly, the solution optimization algorithm is used at the end of each iteration to de-duplicate and optimally reconstruct the current search result. Experiments show that the CTA-kroutes algorithm can effectively improve the equalization of multi-vehicle paths and its applicability to large-scale road networks.

Revisiting the complexity of and algorithms for the graph traversal edit distance and its variants

The graph traversal edit distance (GTED), introduced by Ebrahimpour Boroojeny et al. (2018), is an elegant distance measure defined as the minimum edit distance between strings reconstructed from Eulerian trails in two edge-labeled graphs. GTED can be used to infer evolutionary relationships between species by comparing de Bruijn graphs directly without the computationally costly and error-prone process of genome assembly. Ebrahimpour Boroojeny et al. (2018) propose two ILP formulations for GTED and claim that GTED is polynomially solvable because the linear programming relaxation of one of the ILPs always yields optimal integer solutions. The claim that GTED is polynomially solvable is contradictory to the complexity results of existing string-to-graph matching problems. We resolve this conflict in complexity results by proving that GTED is NP-complete and showing that the ILPs proposed by Ebrahimpour Boroojeny et al. do not solve GTED but instead solve for a lower bound of GTED and are not solvable in polynomial time. In addition, we provide the first two, correct ILP formulations of GTED and evaluate their empirical efficiency. These results provide solid algorithmic foundations for comparing genome graphs and point to the direction of heuristics. The source code to reproduce experimental results is available at https://github.com/Kingsford-Group/gtednewilp/.

Partially broken orientations of Eulerian graphs on closed surfaces

In this paper, we discuss orientations given to edges of Eulerian graphs embedded on closed surfaces, and give a characterization of such a graph having a good orientation, i.e., incoming edges and outgoing edges appear alternately around each vertex of the graph. Furthermore, we consider the partially broken orientations of Eulerian graphs embedded on closed surfaces, that is, one in which only for each of specified vertices, incoming edges and outgoing edges appear alternately around it. We discuss a little about good orientations defined for Eulerian graphs drawn on closed surfaces with crossings, and for general connected graphs, i.e., not necessarily Eulerian graphs, embedded on closed surfaces.

NP-completeness of the Eulerian walk problem for a multiple graph

In this paper, we study undirected multiple graphs of any natural multiplicity $k>1$. There are edges of three types: ordinary edges, multiple edges and multi-edges. Each edge of the last two types is a union of $k$ linked edges, which connect 2 or $(k+1)$ vertices, correspondingly. The linked edges should be used simultaneously. If a vertex is incident to a multiple edge, it can be also incident to other multiple edges and it can be the common end of $k$ linked edges of some multi-edge. If a vertex is the common end of some multi-edge, it cannot be the common end of another multi-edge. We study the problem of finding the Eulerian walk (the cycle or the trail) in a multiple graph, which generalizes the classical problem for an ordinary graph. We prove that the recognition variant of the multiple eulerian walk problem is NP-complete. For this purpose we first prove NP-completeness of the auxiliary problem of recognising the covering trails with given endpoints in an ordinary graph.

Circular flows in mono‐directed signed graphs

AbstractIn this paper, the concept of circular ‐flow in a mono‐directed signed graph is introduced. That is a pair , where is an orientation on and satisfies that for each positive edge and for each negative edge , and the total in‐flow equals the total out‐flow at each vertex. This is the dual notion of circular colorings of signed graphs and is distinct from the concept of circular flows in bi‐directed graphs associated with signed graphs studied in the literature. We first explore the connection between circular ‐flows and modulo ‐orientations in signed graphs. Then we focus on the upper bounds for the circular flow indices of signed graphs in terms of the edge‐connectivity, where the circular flow index of a signed graph is the minimum value such that it admits a circular ‐flow. We prove that every 3‐edge‐connected signed graph admits a circular 6‐flow and every 4‐edge‐connected signed graph admits a circular 4‐flow. More generally, for , we show that every ‐edge‐connected signed graph admits a circular ‐flow, every ‐edge‐connected signed graph has a circular ‐flow with , and every ‐edge‐connected signed graph admits a circular ‐flow. Moreover, the ‐edge‐connectivity condition is shown to be sufficient for a signed Eulerian graph to admit a circular ‐flow, and applying this result to planar graphs, we conclude that every signed bipartite planar graph of negative girth at least admits a homomorphism to the negative even cycles .

Analysis of City Transportation Routes with Fleury Algorithm on Bus Route Network: Case Study of Trans Mamminasata Makassar City Indonesia

Graph theory is a branch of mathematics that studies the structure and describes the relationships between vertices and edges. In general, graph theory is used to represent discrete objects (vertices) and the relationships between them (edges). A path that can pass through each edge exactly once in a graph is called a directed graph Euler path. One way to find Euler paths is by using Fleury's algorithm. Fleury's algorithm is designed to find Euler paths in directed graphs. This article examines the application of Fleury's algorithm to the determination of a transportation route in a city interpreted in a directed graph. The case study in this research focuses on the trans Mamminasata bus route in Makassar city Indonesia with the aim of implementing Fleury's algorithm in bus route generation. The result obtained from the simulation using Fleury's algorithm is that all edges can be visited exactly once, so that an Euler path is formed on the transportation route. The route formed from the Euler trajectory will be a comparison of the current route to determine the operational efficiency of the bus route network in the city.

Compatible Spanning Circuits and Forbidden Induced Subgraphs

A compatible spanning circuit in an edge-colored graph G (not necessarily properly) is defined as a closed trail containing all vertices of G in which any two consecutively traversed edges have distinct colors. The existence of extremal compatible spanning circuits (i.e., compatible Hamilton cycles and compatible Euler tours) has been studied extensively. Recently, sufficient conditions for the existence of compatible spanning circuits visiting each vertex at least a specified number of times in specific edge-colored graphs satisfying certain degree conditions have been established. In this paper, we continue the research on sufficient conditions for the existence of such compatible s-panning circuits. We consider edge-colored graphs containing no certain forbidden induced subgraphs. As applications, we also consider the existence of such compatible spanning circuits in edge-colored graphs G with κ(G) ≥ α(G), κ(G) ≥ α(G) − 1 and κ (G) ≥ α(G), respectively. In this context, κ(G), α(G) and κ (G) denote the connectivity, the independence number and the edge connectivity of a graph G, respectively.

Eulerian 2-Complexes

A famous theorem in graph theory—originating with Euler—characterizes connected even-degree graphs as (1) those graphs that admit an Euler tour, and (2) those connected graphs that decompose as a face-disjoint union of cycles. We explore a 2-dimensional generalization of this theorem, with graphs (i.e., 1-complexes) replaced by 2-complexes. This entails an interesting generalization of cycles, and the introduction of the notion of a “2-dimensional Euler tour.”

A new application of fractional glucose-insulin model and numerical solutions

Along with the developing technology, obesity and diabetes are increasing rapidly among people. The identification of diabetes diseases, modeling, predicting their behavior, conducting simulations, studying control and treatment methods using mathematical methods has become of great importance. In this paper, we have obtained numerical solutions by considering the glucose-insulin fractional model. This model consists of three compartments: the blood glucose concentration (G), the blood insulin concentration (I) and the ready-to-absorb glucose concentration (D) in the small intestine. The fractional derivative is used in the sense of Caputo. By performing mathematical analyzes for the Glucose-Insulin fractional mathematical model, numerical results were obtained with the help of the Euler method and graphs were drawn.

Eulerian Spaces

We develop a unified theory of Eulerian spaces by combining the combinatorial theory of infinite, locally finite Eulerian graphs as introduced by Diestel and Kühn with the topological theory of Eulerian continua defined as irreducible images of the circle, as proposed by Bula, Nikiel and Tymchatyn. First, we clarify the notion of an Eulerian space and establish that all competing definitions in the literature are in fact equivalent. Next, responding to an unsolved problem of Treybig and Ward from 1981, we formulate a combinatorial conjecture for characterising the Eulerian spaces, in a manner that naturally extends the characterisation for finite Eulerian graphs. Finally, we present far-reaching results in support of our conjecture which together subsume and extend all known results about the Eulerianity of infinite graphs and continua to date. In particular, we characterise all one-dimensional Eulerian spaces.

Properties of Graph Based on Divisor-Euler Functions

Divisor function gives the residues of which divide it. A function denoted by counts the total possible divisors of and gives the list of co-prime integers to . Many graphs had been constructed over these arithmetic functions. Using and , a well known graph named as divisor Euler function graph has been constructed. In this paper, we use divisor function and anti Euler function . We label the symbol to count those residues of which are not co-prime to . By using these functions, we find a new graph, called divisor anti-Euler function graph (DAEFG), denoted as . Let be a DAEFG, where and . The objective of this sequel is to introduce and discuss the properties of DAEFG. In this work, we discuss novel classes of proposed graph with its structure using loops, cycles, components of graph, degree of its vertices, components as complete, bipartite, planar, Hamiltonian and Eulerian graphs. Also, we find chromatic number, chromatic index and clique of these graphs.

Voxel-based variable width continuous spiral path planning for 3D printing

Discontinuous printing paths in 3D printing lead to wasted time in fabricating products. This paper proposes a voxel-based continuous spiral path planning algorithm for 3D printing parts with less time consumption. Interestingly, we found our method can also improve the surface qualities and mechanical properties of the printed parts. The proposed path planning strategy in this paper is based on voxels which is different from traditional path planning strategies in literature. Firstly, our proposed algorithm determines the voxel size by considering the printable width of the printer nozzle. It then generates discrete elements by voxelizing the model and creates spirals and line segments by restricting the print direction of the connectable discrete elements. Next, the algorithm employs the Euler circuit to generate continuous sub-paths for the spirals and line segments. It utilizes the depth-first traversal algorithm to traverse and open the continuous sub-paths, resulting in printable continuous paths. To maintain surface accuracy for complex models, contour lines are introduced on the model's exterior and connected internally to ensure global path continuity. Lastly, the algorithm controls the path width in different directions by adjusting the extrusion rate of the printhead, preventing path overlap. Experimental results demonstrate that the proposed algorithm outperforms traditional path planning algorithms and the Connected Fermat spirals algorithm. The proposed strategy effectively mitigates the negative effects of path discontinuity, reduces printing time, and improves printed accuracy, and mechanical properties.

The algorithms for the Eulerian cycle and Eulerian trail problems for a multiple graph

In this paper, we study undirected multiple graphs of any natural multiplicity $k>1$. There are edges of three types: ordinary edges, multiple edges and multi-edges. Each edge of the last two types is a union of $k$ linked edges, which connect 2 or $(k+1)$ vertices, correspondingly. The linked edges should be used simultaneously. If a vertex is incident to a multiple edge, it can be also incident to other multiple edges and it can be the common end of $k$ linked edges of some multi-edge. If a vertex is the common end of some multi-edge, it cannot be the common end of another multi-edge. We set the problem of finding the eulerian walk (the cycle or the trail) in a multiple graph, which generalizes the classical problem for an ordinary graph. We formulate the necessary conditions for existence of an eulerian walk in a multiple graph and show that these conditions are not sufficient. Besides that, we show that the necessary conditions of existence of an eulerian cycle and eulerian trail are not mutually exclusive for an arbitrary multiple graph, that is why it is possible to construct a multiple graph where two types of eulerian walks exist simultaneously. Any multiple graph can be juxtaposed to the ordinary graph with quasi-vertices, which represents the structure of the initial graph in a simpler form. In particular, each eulerian walk in the multiple graph corresponds to the eulerian walk in the graph with quasi-vertices. The algorithm for getting such a graph is formulated. Also, the auxiliary problem of finding the covering trails with given endpoints in an ordinary graph is studied. Two algorithms are obtained for this problem. We elaborate the algorithm for finding the eulerian walk in a multiple graph, which has the exponential complexity. We suggest the polynomial algorithm for the special case of a multiple graph and show that the necessary conditions are sufficient for existence of an eulerian walk in this special case.

Euler dynamic H-trails in edge-colored graphs

Alternating Euler trails has been extensively studied for its diverse practical and theoretical applications. Let H be a graph possibly with loops and G be a multigraph without loops. In this paper we deal with any fixed coloration of E(G) with V(H) (H-coloring of G). A sequence W = ( v 0 , e 0 1 , … , e 0 k 0 , v 1 , e 1 1 , … , e n − 1 k n − 1 , v n ) in G, where for each i ∈ { 0 , … , n − 1 } , k i ≥ 1 and e i j = v i v i + 1 is an edge in G, for every j ∈ { 1 , … , k i } , is a dynamic H-trail if W does not repeat edges and c ( e i k i ) c ( e i + 1 1 ) is an edge in H, for each i ∈ { 0 , … , n − 2 } . In particular, a dynamic H-trail is an alternating trail when H is a complete graph without loops and ki = 1, for every i ∈ { 1 , … , n − 1 } . In this paper, we introduce the concept of dynamic H-trail, which arises in a natural way in the modeling of many practical problems, in particular, in theoretical computer science. We provide necessary and sufficient conditions for the existence of closed Euler dynamic H-trail in H-colored multigraphs. Also we provide polynomial time algorithms that allows us to convert a cycle in an auxiliary graph, L 2 H ( G ) , in a closed dynamic H-trail in G, and vice versa, where L 2 H ( G ) is a non-colored simple graph obtained from G in a polynomial time.