Eulerian Graphs Research Articles

Eulerian Walks in Temporal Graphs

An Eulerian walk (or Eulerian trail) is a walk (resp. trail) that visits every edge of a graph G at least (resp. exactly) once. This notion was first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. But what if Euler had to take a bus? In a temporal graph varvec{(G,lambda )}, with varvec{lambda : E(G)}varvec{rightarrow } varvec{2}^{varvec{[tau ]}}, an edge varvec{e}varvec{in } varvec{E(G)} is available only at the times specified by varvec{lambda (e)}varvec{subseteq } varvec{[tau ]}, in the same way the connections of the public transportation network of a city or of sightseeing tours are available only at scheduled times. In this paper, we deal with temporal walks, local trails, and trails, respectively referring to edge traversal with no constraints, constrained to not repeating the same edge in a single timestamp, and constrained to never repeating the same edge throughout the entire traversal. We show that, if the edges are always available, then deciding whether varvec{(G,lambda )} has a temporal walk or trail is polynomial, while deciding whether it has a local trail is varvec{texttt {NP}}-complete even if varvec{tau = 2}. In contrast, in the general case, solving any of these problems is varvec{texttt {NP}}-complete, even under very strict hypotheses. We finally give varvec{texttt {XP}} algorithms parametrized by varvec{tau } for walks, and by varvec{tau +tw(G)} for trails and local trails, where varvec{tw(G)} refers to the treewidth of varvec{G}.

Edge Exploration of Temporal Graphs

We introduce a natural temporal analogue of Eulerian circuits and prove that, in contrast to the static case, it is {textsc {NP}}-hard to determine whether a given temporal graph is temporally Eulerian even if strong restrictions are placed on the structure of the underlying graph and each edge is active at only three times. However, we do obtain an {textsc {FPT}}-algorithm with respect to a new parameter called interval-membership-width which restricts the times assigned to different edges; we believe that this parameter will be of independent interest for other temporal graph problems. Our techniques also allow us to resolve two open questions of Akrida, Mertzios and Spirakis [CIAC 2019] concerning a related problem of exploring temporal stars.

Graph Theory: A Lost Component For Development in Nigeria

Graph theory is one of the neglected branches of mathematics in Nigeria but with the most applications in other fields of research. This article shows the paucity, importance, and necessity of graph theory in the development of Nigeria. The adjacency matrix and dual graph of the Nigeria map were presented. The graph spectrum and energies (graph energy and Laplacian energy) of the dual graph were computed. Then the chromatic number, maximum degree, minimum spanning tree, graph radius, and diameter, the Eulerian circuit and Hamiltonian paths from the dual graph were obtained and discussed.

Even Cycle Decompositions of index 3 by a novel coloring technique

An even cycle decomposition of an Eulerian graph is a partition of the edge set into even cycles. We color the even cycles so as two cycles sharing at least one vertex receive distinct colors. If k is the minimum number of required colors in such a coloring, then the even cycle decomposition has index k . We prove that the line graph of every bridgeless cubic graph of oddness 2 has an even cycle decomposition of index 3. The same property holds for the line graphs of some infinite families of class 2 cubic graphs with arbitrary large oddness. The construction of even cycle decompositions of index 3 in the line graph of a class 2 cubic graph is alternative to the constructions that are known in the literature; that one for the line graph of a cubic graph with arbitrary large oddness is also a new contribution to the more general problem on the existence of even cycle decompositions in the line graph of a bridgeless cubic graph. The constructions are obtained by applying a novel coloring technique on the edges of the line graph.

Enumeration of Triangles and Hamiltonian Property of The Zero-Divisor Cayley Graph of The Ring G(Zₙ,⊕,⊙)

In this paper an enumeration method to find the number of triangles in the zero-divisor Cayley graph G(Zₙ,D₀ ) associated with the ring (Zₙ,⨁,⨀),n≥1 of integers modulo n, an integer and its subset D0 of zero-divisors is presented. Further it is shown that this graph is Hamiltonian, not bipartite and Eulerian graph when n is odd.

A novel single-stroke path planning algorithm for 3D printers using continuous carbon fiber reinforced thermoplastics

Recently, 3D printers using continuous carbon fiber as a material have attracted considerable attention from researchers due to its high mechanical properties and the potential to reproduce complex shapes. Although the main advantage of 3D printing is that complex shapes can be produced through a simple process, very few studies have evaluated samples with complex shapes containing continuous fibers. Unlike pure resin or materials mixed with short fibers, carbon fiber filament contains long, strong fibers, which requires additional steps such as cutting the filament multiple times to create complex shapes. This study proposes a path planning algorithm for the print head of a fused deposition modeling (FDM) 3D printer using filaments reinforced with continuous fiber bundle. The proposed method generates a single-stroke path that can be printed continuously without cutting the filament for each layer. Specifically, an Eulerian graph is created from the print pattern, and the path is generated using Hierholzer’s Algorithm with constraints. By visualizing the generated paths, we confirmed that the paths generated by the proposed method do not interfere with each other. Furthermore, samples were fabricated by using a 3D printer capable of printing with continuous fibers, and the inside of these samples were observed with X-ray computed tomography (CT). In addition, the gaps that occur during actual printing can be suppressed by adjusting the parameters of the algorithm. Finally, we have shown that the proposed method can be applied to geometries with varying wall thicknesses and aperiodic patterns using the Voronoi diagram.

THE USE OF MARKOV ALGORITHMS FOR THE STUDY OF l-VOIDS IN BIG DATA OF SOCIO-ECONOMIC SYSTEMS. PART 2

The second part of the article is presented. Big data l-voids are considered using the N-scheme of the Markov algorithm. The diagrams of occurrences of l-voids in a semi-Eulerian cycle containing an Euler path, a matroid and an incomplete Fano matroid, minors K3, 3 and K5, an extra large cycle of occurrences are analyzed. An example of reconstructing a fragment of an incomplete Fano matroid with l-voids is considered. Examples are given for independent implementation of the method of filling the artificial intelligence database (AnwM f typeK) DB based on the results of the analysis of l-voids in the N-scheme of the Markov algorithm. The database is populated using a formal language with the alphabet M = litj/abdgckm where litj are terminal elements, abdgckm are non-terminal elements, and the basis of the alphabet is i – initial, t – terminal, j – isomorphic, l-empty morphisms. The p – initial, e – simple, and h – final occurrences of words from the Markov alphabet A are determined. An example of a simplified numbering of occurrence chains is given, which ensures the selection of all objects represented by words in the Markov alphabet A, belonging to “their own” chain of occurrences, as well as all related to chain object. The mechanism of dynamic change of the number-type in the N-scheme of the Markov algorithm is presented, which provides the analysis of the word as an evolving category, as well as the evolution of categories, which includes such a word. It has been established that normal Markov inference is characterized by temporal conditionality and locality. The conclusion is made about the expediency of considering the N-scheme in a distributed computing system. In this case, it is necessary to analyze the diagrams of occurrences of objects that have several designations with words from the Markov alphabet A, in order to identify them and process data from such objects.

Skeleton Curve-Guided Five-Axis Sweep Scanning for Surface With Multiple Holes

Although surface with multiple holes (SMHs) is widely used in industrial components, the precise inspection of SMH is a knotty task due to the inefficiency of traditional surface inspection technology. Recently, a five-axis surface sweep scanning approach is emerging and shows great potential to boost the surface inspection efficiency by sensing surface in a continuous sweep scanning way. However, an efficient sweep scanning path should cater to the unique kinematic characteristics of the five-axis inspection system. In this article, we present an approach to automatically generate efficient five-axis sweep scanning paths for inspecting SMH. First, the skeleton of SMH is extracted along with the surface partitioned into several patches, based on which a guiding curve-based sweep scanning path can be defined for each surface patch. By modeling the skeleton of SMH as a directed Euler graph and finding a proper sequence to traverse the graph, a continuous five-axis sweep scanning path is then generated to sweep all patches without any transition among them. The nonsweeping time could be eliminated so that the sweep scanning efficiency is drastically improved in this way. Simulation and physical inspection experiments are conducted on two SMHs, showing that our method significantly outperforms existing approaches, such as the popular zigzag method and the method from the leading commercial software of RENISHAW. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This article aims to generate an efficient sweep scanning path of a five-axis coordinate measuring machine (CMM) for inspecting the SMHs of a precision component, such as an engine block or a gearbox cover. Although traditional five-axis CMM path generation methods can be applied to SMH inspection, most of them are designed for the point-by-point inspection mode, which intrinsically suffers from extremely low efficiency. Recently, an emerging five-axis surface sweep scanning technology shows great potential for boosting the efficiency of surface inspection by using a continuous surface sensing stylus mounted on a rapid rotary probe head. The realization of high-speed five-axis surface scanning depends on a proper sweep scanning path which takes advantage of the superior kinematic performance of the rotary probe head as much as possible and accordingly avoids assigning heavy workload to the three linear axes with weaker kinematic performance. However, up to date, only a handful of five-axis sweep scanning path planning methods are reported and mainly focus on the simple and compact surface. Therefore, existing methods are incompetent to generate a suitable sweep scanning path for high-speed inspection of SMH. For these reasons, we present a novel and practical approach to automatically generate a continuous five-axis sweep scanning path for complex SMH. The generated path exactly meets the unique kinematic requirements of high-speed five-axis inspection and eliminates all noninspection operations. In this way, the SMH inspection can be accomplished with high efficiency.

THx2 Programmable Logic Block Architecture for Clockless Asynchronous FPGAs

To address some of the challenges of asynchronous design, we propose a new, decomposable asynchronous logic block architecture based on our TH<inline-formula> <tex-math notation="LaTeX">$x2$ </tex-math></inline-formula> programmable threshold cell, and we use it to implement common threshold functions found in asynchronous, null convention logic circuits. At a minimum, programmable gate arrays require a programmable logic cell that can implement a complete set of logic. It is well known that a NAND function forms a complete set of logic, and in null convention logic, the TH12 and TH22 threshold cells are used to form a basic two-input NAND function. The TH<inline-formula> <tex-math notation="LaTeX">$x2$ </tex-math></inline-formula> threshold cell is capable of performing both TH12 and TH22 operations, so it too forms a complete set of logic. In this paper, we present our eight-transistor mask-programmable gate array logic cell, 16-transistor field-programmable gate array logic cell, and new decomposable field-programmable gate array logic block architecture, all based on the TH<inline-formula> <tex-math notation="LaTeX">$x2$ </tex-math></inline-formula> threshold cell and suitable for implementing null convention logic asynchronous functions. To minimize the TH<inline-formula> <tex-math notation="LaTeX">$x2$ </tex-math></inline-formula> threshold cell area for both TH12 and TH22 modes, we designed a layout with common Euler paths and no diffusion breaks for both modes. The highly compact nature of the TH<inline-formula> <tex-math notation="LaTeX">$x2$ </tex-math></inline-formula> threshold cell&#x2013;along with the symmetry of the mask- and field-programmable gate array logic cells&#x2013;made it an ideal candidate for an asynchronous field-programmable logic block structure. This paper is part of an ongoing project, and it only addresses the programmable logic block architecture, not a complete FPGA fabric.

The graphs with a symmetrical Euler cycle

The graphs in this paper are finite, undirected, and without loops, but may have more than one edge between a pair of vertices. If such a graph has ℓ edges, then an Euler cycle is a sequence (e1,e2,…,eℓ) of these ℓ edges, each occurring exactly once, such that ei, ei + 1 are incident with a common vertex for each i (reading subscripts modulo ℓ). An Euler cycle is symmetrical if there exists an automorphism of the graph such that ei → ei + 2 for each i. The cyclic group generated by this automorphism has one orbit on edges if ℓ is odd, or two orbits of length ℓ/2 if ℓ is even: that is to say, the group is regular or bi-regular on edges, respectively. Symmetrical Euler cycles arise naturally from arc-transitive embeddings of graphs in surfaces since, for each face of the embedded graph, the sequence of edges on the boundary of the face forms a symmetrical Euler cycle for the induced subgraph on this edge-set. We first classify all finite connected graphs which admit a cyclic subgroup of automorphisms that is regular or bi-regular on edges, and identify more than a dozen infinite families of examples. We then prove that exactly six of these families consist of graphs with symmetrical Euler cycles. These are the (only) candidates for the induced subgraphs of the boundary cycles of the faces of arc-transitive maps.

Dense Eulerian Graphs are $(1, 3)$-Choosable

A graph $G$ is total weight $(k,k')$-choosable if for any total list assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ real numbers, and each edge $e$ a set $L(e)$ of $k'$ real numbers, there is a proper total $L$-weighting, i.e., a mapping $f: V(G) \cup E(G) \to \mathbb{R}$ such that for each $z \in V(G) \cup E(G)$, $f(z) \in L(z)$, and for each edge $uv$ of $G$, $\sum_{e \in E(u)}f(e)+f(u) \ne \sum_{e \in E(v)}f(e) + f(v)$. This paper proves that if $G$ decomposes into complete graphs of odd order, then $G$ is total weight $(1,3)$-choosable. As a consequence, every Eulerian graph $G$ of large order and with minimum degree at least $0.91|V(G)|$ is total weight $(1,3)$-choosable. We also prove that any graph $G$ with minimum degree at least $0.999|V(G)|$ is total weight $(1,4)$-choosable.

THE USE OF MARKOV ALGORITHMS FOR THE STUDY OF l-VOIDS IN BIG DATA OF SOCIO-ECONOMIC SYSTEMS. Part 1

A synthesized N-scheme of the Markov algorithm is considered, which can be replaced by his wellknown γ-scheme. Using the N-scheme of the Markov algorithm, the properties of l-voids of big data of socio-economic systems are investigated as an element of the typology of the risk of money laundering. In the N-scheme, the numbers-types w in the alphabet M = litj/abdgckm of words in the Markov alphabet A that make up the l-voids are determined. The alphabet M is considered as a formal grammar consisting of terminal litj and non-terminal abdgckm symbols. Examples of l-voids in the form of a simple square cycle and an incomplete Euler cycle are considered. Examples of filling in the artificial intelligence database for l-voids are given in order to conduct an independent experiment on checking the N-scheme of the Markov algorithm. It is proposed to represent l-voids of big data of socio-economic systems in the form of a diagram of occurrences of words in the Markov alphabet А and subject them to analysis. The analysis of occurrence diagrams is proposed to be carried out using the N-scheme of the Markov algorithm. In addition, we present the results of the analysis of the l-voids of a simple square cycle with heterogeneous data, as well as the analysis of its locally lh-final occurrences and under conditions where it is impossible to verify morphisms in time. It is concluded that the study of methods for applying hashing in conjunction with the N-scheme of the Markov algorithm, in order to determine the most appropriate hash function and algorithm for its implementation, is a promising scientific task. It is expedient to synthesize a specialized data container using hash functions and hash tables for the N-scheme of the Markov algorithm. It is indicated that for verification, it is possible to implement the N-scheme of the Markov algorithm using hash functions and hash tables known in modern programming languages.

A-trails of embedded graphs and twisted duals

Kotzig showed that every connected 4 -regular plane graph has an A -trail—an Eulerian circuit that turns either left or right at each vertex. However, this statement is not true for Eulerian plane graphs and determining if an Eulerian plane graph has an A -trail is NP-hard. The aim of this paper is to give a characterization of Eulerian embedded graphs having an A -trail. Andersen et al. showed the existence of orthogonal pairs of A -trails in checkerboard colourable 4 -regular graphs embedded on the plane, torus and projective plane. A problem posed in their paper is to characterize Eulerian embedded graphs (not necessarily checkerboard colourable) which contain two orthogonal A -trails. In this article, we solve this problem in terms of twisted duals. Several related results are also obtained.

Constant amortized time enumeration of Eulerian trails

In this paper, we consider enumeration problems for edge-distinct and vertex-distinct Eulerian trails. Two Eulerian trails are said to be edge-distinct if the edge sequences are not identical, and they are said to be vertex-distinct if the vertex sequences are not identical. To solve these problems, we propose optimal enumeration algorithms that run in O(N+m) total time, where N is the number of solutions and m is the number of edges in an input connected graph. The proposed algorithms are based on the reverse search technique introduced by [Avis and Fukuda, DAM 1996], and the push-out amortization technique introduced by [Uno, WADS 2015].

Z-oriented triangulations of surfaces

The main objects of the paper are z -oriented triangulations of connected closed 2 -dimensional surfaces. A z -orientation of a map is a minimal collection of zigzags which double covers the set of edges. We have two possibilities for an edge – zigzags from the z -orientation pass through this edge in different directions (type I) or in the same direction (type II). Then there are two types of faces in a triangulation: the first type is when two edges of the face are of type I and one edge is of type II and the second type is when all edges of the face are of type II. We investigate z -oriented triangulations with all faces of the first type (in the general case, any z -oriented triangulation can be shredded to a z -oriented triangulation of such type). A zigzag is homogeneous if it contains precisely two edges of type I after any edge of type II. We give a topological characterization of the homogeneity of zigzags; in particular, we describe a one-to-one correspondence between z -oriented triangulations with homogeneous zigzags and closed 2 -cell embeddings of directed Eulerian graphs in surfaces. At the end, we give an application to one type of the z -monodromy.

Covering hypergraphs are Eulerian

An Euler tour in a hypergraph (also called a rank-2 universal cycle or 1-overlap cycle in the context of designs) is a closed walk that traverses every edge exactly once. In this paper, we define a covering $k$-hypergraph to be a non-empty $k$-uniform hypergraph in which every $(k-1)$-subset of vertices appear together in at least one edge. We then show that every covering $k$-hypergraph, for $k\geq 3$, admits an Euler tour if and only if it has at least two edges.

A note on Hamiltonian cycles in planar graphs

Unlike the problem of finding Eulerian cycles, the problem of deciding whether a given graph has a Hamiltonian cycle is NP-complete, even when restricting to planar graphs. There are however some criteria that can be used in special cases. One of them can be derived by Grinberg’s theorem. In this note, we give some more (slightly more general) criteria derived from Grinberg’s theorem to show that a graph does not have a Hamiltonian cycle.

A General Branch-and-Cut Framework for Rotating Workforce Scheduling

In this paper, we propose a general algorithmic framework for rotating workforce scheduling. We develop a graph representation that allows to model a schedule as a Eulerian cycle of stints, which we then use to derive a problem formulation that is compact toward the number of employees. We develop a general branch-and-cut framework that solves rotating workforce scheduling in its basic variant, as well as several additional problem variants that are relevant in practice. These variants comprise, among others, objectives for the maximization of free weekends and the minimization of employees. Our computational studies show that the developed framework constitutes a new state of the art for rotating workforce scheduling. For the first time, we solve all 6,000 instances of the status quo benchmark for rotating workforce scheduling to optimality with an average computational time of 0.07 seconds and a maximum computational time of 2.53 seconds. These results reduce average computational times by more than 99% compared with existing methods. Our algorithmic framework shows consistent computational performance, which is robust across all studied problem variants. Summary of Contribution: This paper proposes a novel exact algorithmic framework for the well-known rotating workforce scheduling problem (RWSP). Although the RWSP has been extensively studied in different problem variants and for different exact and heuristic solution approaches, the presented algorithmic framework constitutes a new state-of-the-art for the RWSP that solves all known benchmark sets to optimality and improves on the current state-of-the-art by orders of magnitude with respect to computational times, especially for large-scale instances. The paper is both of methodological value for researchers and of high interest for practitioners. For researchers, the presented framework is amenable for various problem variants and provides a common ground for further studies and research. For practitioners and software developers, low computational times of a few seconds allows the framework to be to embedded into personnel scheduling software.

A 2-connected graph in two dimensions containing 25 vertices with Gallai's Property

Graph theory recently considers as a modern field of mathematics and it was introduced by a great mathematician Leonhard Euler in 1735. Late then it is a flowered in the strong tool used in closely each area of science and nowadays it is most attractive and active area of mathematics research. Actually, graph theory is the study of graph, structure and molecules which have association in each other. It is very essential part of discrete mathematics which has been created with collaboration of nodes (vertices) and edges (lines or links), if a graph is connected with vertices directly by edges it is direct graph or a graph having symmetrically lines is also called direct graph or diagraph. In the studies of graph, we have investigated so many kinds of graph, Hypo-Hamiltonian is one of them. It is most important and it can be defined as the graph which hasn't Hamiltonian cycle but can be developed with the removing of single vertex form the developed graph. Lot of researchers have given the contribution in this research just like Naeem et al has worked on "A Two-Connected Graph with Gallai's Property" containing 12,18 and 25 vertices. In this research paper we have developed two different graphs consisting 25 Eulerian graph which has highest cycle and path order C( G) — 24 and P( G) — 25. Other graph is in 3D under cycle which contains 23 nodes.

A circle criterion for a generalized cross graph in terms of minimal excluded minors

Geelen and Oum described classes of minimal excluded pivot-minors for a simple graph to be a circle graph and for a delta-matroid to be Eulerian. Pivot-equivalence classes of circle simple graphs and delta-matroids arise in the investigation of Eulerian cycles on cross graphs (4-valent graphs with cross structure). The results established by Geelen and Oum rely on some lemmas in their work, which are shown below to be not quite correct. We consider generalized cross graphs, which arise in the description of rotating circuits on cross graphs. For such graphs we derive a circle criterion: we reproduce and augment the arguments due to Geelen and Oum, and we improve some incorrectly formulated statements. As a result, we obtain the same list of 166 inequivalent graphs, the minimal excluded minors for a generalized cross graph to be a circle graph. Bibliography: 14 titles.