Eulerian Graphs Research Articles

What do Eulerian and Hamiltonian cycles have to do with genome assembly?

Many students are taught about genome assembly using the dichotomy between the complexity of finding Eulerian and Hamiltonian cycles (easy versus hard, respectively). This dichotomy is sometimes used to motivate the use of de Bruijn graphs in practice. In this paper, we explain that while de Bruijn graphs have indeed been very useful, the reason has nothing to do with the complexity of the Hamiltonian and Eulerian cycle problems. We give 2 arguments. The first is that a genome reconstruction is never unique and hence an algorithm for finding Eulerian or Hamiltonian cycles is not part of any assembly algorithm used in practice. The second is that even if an arbitrary genome reconstruction was desired, one could do so in linear time in both the Eulerian and Hamiltonian paradigms.

Path and cycle decompositions of dense graphs

We make progress on three long standing conjectures from the 1960s about path and cycle decompositions of graphs. Gallai conjectured that any connected graph on $n$ vertices can be decomposed into at most $\left\lceil \frac{n}{2}\right\rceil$ paths, while a conjecture of Haj\'{o}s states that any Eulerian graph on $n$ vertices can be decomposed into at most $\left\lfloor \frac{n-1}{2}\right\rfloor$ cycles. The Erd\H{o}s-Gallai conjecture states that any graph on $n$ vertices can be decomposed into $O(n)$ cycles and edges. We show that if $G$ is a sufficiently large graph on $n$ vertices with linear minimum degree, then the following hold. (i) $G$ can be decomposed into at most $\frac{n}{2}+o(n)$ paths. (ii) If $G$ is Eulerian, then it can be decomposed into at most $\frac{n}{2}+o(n)$ cycles. (iii) $G$ can be decomposed into at most $\frac{3 n}{2}+o(n)$ cycles and edges. If in addition $G$ satisfies a weak expansion property, we asymptotically determine the required number of paths/cycles for each such $G$. (iv) $G$ can be decomposed into $\max \left\{\frac{odd(G)}{2},\frac{\Delta(G)}{2}\right\}+o(n)$ paths, where $odd(G)$ is the number of odd-degree vertices of $G$. (v) If $G$ is Eulerian, then it can be decomposed into $\frac{\Delta(G)}{2}+o(n)$ cycles. All bounds in (i)-(v) are asymptotically best possible.

Optimization of stochastic chinese postman networks problem using priorities

The postman problem is concern with the choice of root to follow in order that he or she pass each streets at least once and return at starting point with least traveling distance. The Chinese postman problem is transformed as network optimization problem under certain constraint. Stochastic networks are networks of non-binary vertices that differ over time, reflecting the probability of a connection of two nodes. This paper deals with theoretical and mathematical foundation of stochastic networks related to an additional feature of the Chinese postman Problem (CPP) is that postman visited with first priority node as soon as possible. Fleury’s and Dijkstra’s algorithm have been used with respect to Eulerian path. Finally, this paper also illustrates and justify the implementation of the methodology and algorithms. Consideration of cost factors in the ever network problems gives another scope of research.

Modeling and Classification of Alluvial Fans with DEMs and Machine Learning Methods: A Case Study of Slovenian Torrential Fans

Alluvial (torrential) fans, especially those created from debris-flow activity, often endanger built environments and human life. It is well known that these kinds of territories where human activities are favored are characterized by increasing instability and related hydrological risk; therefore, treating the problem of its assessment and management is becoming strongly relevant. The aim of this study was to analyze and model the geomorphological aspects and the physical processes of alluvial fans in relation to the environmental characteristics of the territory for classification and prediction purposes. The main geomorphometric parameters capable of describing complex properties, such as relative fan position depending on the neighborhood, which can affect their formation or shape, or properties delineating specific parts of fans, were identified and evaluated through digital elevation model (DEM) data. Five machine learning (ML) methods, including a hybrid Euler graph ML method, were compared to analyze the geomorphometric parameters and physical characteristics of alluvial fans. The results obtained in 14 case studies of Slovenian torrential fans, validated with data of the empirical model proposed by Bertrand et al. (2013), confirm the validity of the developed method and the possibility to identify alluvial fans that can be considered as debris-flow prone.

Harary index of Eulerian graphs

The Harary index of a connected (molecular) graph is defined as the sum of reciprocals of distances between all its pairs of vertices. In this paper, we characterize the extremal Eulerian graphs (with a fixed number of vertices) with greatest and smallest Harary indices and characterize the Harary indices of two special type of Eulerian graphs.

A method for efficiently routing order pickers in the leaf warehouse

Warehouse order picking is a critical operation for every supply chain because of its direct influence on customer satisfaction and the high time investment that is usually required for completing it. Hence, improving order picking efficiency can enhance both customer satisfaction and warehouse throughput. The most critical step in order picking is the travelling along the aisles of the warehouse, which accounts for a major share of the total warehouse operating cost. To minimize this cost for the special case of a warehouse with a leaf layout, the paper at hand proposes an efficient optimal order picker routing policy based on an Eulerian graph and a dynamic programming procedure. It further develops four simple routing heuristics, referred to as leaf S-shape, leaf return, leaf midpoint, and leaf largest gap, for guiding the order picker through the leaf warehouse. The performance of the routing heuristics is compared to the exact algorithm proposed in this paper in a systematic numerical experiment. Finally, we present some managerial insights for managing order picking in the leaf warehouse.

Efficient Algorithms for Max-Weighted Point Sweep Coverage on Lines.

As an important application of wireless sensor networks (WSNs), deployment of mobile sensors to periodically monitor (sweep cover) a set of points of interest (PoIs) arises in various applications, such as environmental monitoring and data collection. For a set of PoIs in an Eulerian graph, the point sweep coverage problem of deploying the fewest sensors to periodically cover a set of PoIs is known to be Non-deterministic Polynomial Hard (NP-hard), even if all sensors have the same velocity. In this paper, we consider the problem of finding the set of PoIs on a line periodically covered by a given set of mobile sensors that has the maximum sum of weight. The problem is first proven NP-hard when sensors are with different velocities in this paper. Optimal and approximate solutions are also presented for sensors with the same and different velocities, respectively. For M sensors and N PoIs, the optimal algorithm for the case when sensors are with the same velocity runs in time; our polynomial-time approximation algorithm for the case when sensors have a constant number of velocities achieves approximation ratio ; for the general case of arbitrary velocities, and approximation algorithms are presented, respectively, where integer is the tradeoff factor between time complexity and approximation ratio.

Circuit [formula omitted]-covers of signed graphs

Let G be a signed graph and F a set of signed circuits in G. For an edge e∈E(G), F(e) denotes the number of signed circuits in F that contain e. F is called a circuit-cover of G if F(e)>0 for each e∈E(G), and a circuit k-cover of G if F(e)=k for each e∈E(G). G is coverable if it has a circuit-cover. The existence of a circuit-cover in G is equivalent to the existence of a nowhere-zero flow in G. For a coverable signed graph G, it is proved in this paper that if each maximal 2-edge-connected subgraph of G is eulerian, then G has a circuit 6-cover, consisting of four circuit-covers of G, and as an immediate consequence, G has a circuit-cover of length at most 32|E(G)|, generalizing a known result on signed eulerian graphs. New results on circuit k-covers are obtained and applied to estimating bounds on the lengths of shortest circuit-covers of signed graphs.

Circuit Covers of Signed Eulerian Graphs

A signed circuit cover of a signed graph is a natural analog of a circuit cover of a graph, and is equivalent to a covering of its corresponding signed-graphic matroid with circuits. It was conjectured that a signed graph whose signed-graphic matroid has no coloops has a 6-cover. In this paper, we prove that the conjecture holds for signed Eulerian graphs.

On-Line Test of Pin-Constrained Digital Microfluidic Biochips with Connect-5 Structure

Digital microfluidic biochips (DMFBs) are widely used in the field of biochemistry. Effective off-line and on-line test for the biochips are required to ensure the system reliability. For direct addressing digital microfluidic biochips (DDMFBs), each control pin corresponds to only one electrode, and that can facilitate the testing of such biochips. However, in pin-constrained digital microfluidic biochips (PDMFBs), multiple electrodes may share one control pin, and thus the testing will be more difficult. In this paper, the pin constraint formula for PDMFBs with connect-5 structure is derived. A novel pin assignment scheme is also proposed, which can conduct on-line test that rarely considered by the previous methods. Furthermore, a hybrid method combining the priority strategy and genetic algorithm is introduced for the on-line test of pin-constrained digital microfluidic biochip with connect-5 structure. The simulation results show that the shortest test path acquired by the proposed method is equal to the optimal value of Euler path, which indicates that the method can effectively implement the on-line test of PDMFBs with connect-5 structure.

Incorporating Maintenance Infeasibilities in an Aircraft Rotation Planning Model

In any airline’s schedule development process, aircraft rotations must be planned for individual fleet types after fleet assignment. The aircraft rotation plans must conform to stringent maintenance requirements and this problem can be formulated as a periodic routing problem on an Eulerian graph. We analyze the computational complexity of developing maintenance rotations when some overnighting aircraft may not have sufficient time on the ground to complete extended maintenance (referred to as a maintenance infeasibility). The paper also provides a theoretical analysis of heuristics for the aircraft maintenance rotation problem with maintenance infeasibilities.

Long cycles, heavy cycles and cycle decompositions in digraphs

Hajós conjectured in 1968 that every Eulerian n-vertex graph can be decomposed into at most ⌊(n−1)/2⌋ edge-disjoint cycles. This has been confirmed for some special graph classes, but the general case remains open. In a sequence of papers by Bienia and Meyniel (1986) [1], Dean (1986) [7], and Bollobás and Scott (1996) [2] it was analogously conjectured that every directed Eulerian graph can be decomposed into O(n) cycles.In this paper, we show that every directed Eulerian graph can be decomposed into O(nlog⁡Δ) disjoint cycles, thus making progress towards the conjecture by Bollobás and Scott. Our approach is based on finding heavy cycles in certain edge-weightings of directed graphs. As a further consequence of our techniques, we prove that for every edge-weighted digraph in which every vertex has out-weight at least 1, there exists a cycle with weight at least Ω(log⁡log⁡n/log⁡n), thus resolving a question by Bollobás and Scott.

Codegree conditions for cycle decompositions and Euler tours in 3-uniform hypergraphs

We show that 3-graphs whose codegree is at least (2/3 + o(1))n can be decomposed into tight cycles and admit Euler tours, subject to the trivial necessary divisibility conditions. We also provide a construction showing that our bounds are best possible up to the o(1) term. All together, our results answer in the negative some recent questions of Glock, Joos, Kühn and Osthus.

FIBONACCI AND SUPER FIBONACCI GRACEFUL LABELLINGS OF SOME TYPES OF GRAPHS

We consider the basic theoretical information regarding the Fibonacci graceful graphs. An injective function is said a Fibonacci graceful labelling of a graph of a size , if it induces a bijective function on the set of edges , where by the rule , for any adjacent vertices A graph that allows such labelling is called Fibonacci graceful. In this paper, we introduce the concept of super Fibonacci graceful labelling, narrowing the set of vertex labels, i.e. Four types of problems to be studied are selected. In the problem of the first type, the following question is raised: is there a graph that allows a certain kind of labelling, and under what conditions does this take place? The problem of the second type is the problem of construction: it is necessary, for a given system of requirements for the graph, to construct (at least one) its labelling that would satisfy this system. The following two types of problems relate to enumeration problems: for a given graph, determine the number of different Fibonacci and / or super Fibonacci graceful labellings; build all the different labellings of a given kind. As a result of solving these problems, functions were found that generate Fibonacci and super Fibonacci graceful labellings for graphs of cyclic structure; necessary and sufficient conditions for the existence of Fibonacci graceful labelling for disjunctive union of cycles, super Fibonacci graceful labelling for cycles, Eulerian graphs are obtained; the number of non-equivalent labellings of the cycle is determined; conditions for the existence of a super Fibonacci graceful labelling of a one-point connection of arbitrary connected super Fibonacci graceful graphs … …, are presented

Application of the Fleury Algorithm for Construction of the Euler Cycle on the Cross Stitch Pattern

Application of the Fleury Algorithm for Construction of the Euler Cycle on the Cross Stitch Pattern

Multi-Axis Support-Free Printing of Freeform Parts with Lattice Infill Structures

In additive manufacturing, infill structures are commonly used to reduce the weight and cost of a solid part. Currently, most infill structure generation methods are based on the conventional 2.5-axis printing configuration, which, although able to satisfy the self-supporting condition on the infills, suffer from the well-known stair-case effect on the finished surface and the need of extensive support for overhang features. In this paper, based on the emerging continuous multi-axis printing configuration, we present a new lattice infill structure generation algorithm, which is able to achieve the self-supporting condition for both the infills and the boundary surface of the part. The algorithm critically relies on the use of three mutually orthogonal geodesic distance fields that are embedded in the tetrahedral mesh of the solid model. The intersection between the iso-geodesic distance surfaces of these three geodesic distance fields naturally forms the desired lattice of infill structure, while the density of the infills can be conveniently controlled by adjusting the iso-values. The lattice infill pattern in each curved slicing layer is trimmed to conform to an Eulerian graph so to generate a continuous printing path, which can effectively reduce the retractions of the nozzle during the printing process. In addition, to cater to the collision-free requirement and to improve the printing efficiency, we also propose a printing sequence optimization algorithm for determining a collision-free order of printing of the connected lattice infills, which seeks to reduce the air-move length of the nozzle. Ample experiments in both computer simulation and physical printing are performed, and the results give a preliminary confirmation of the advantages of our methodology.

Deviation estimates for Eulerian edit numbers of random graphs

Consider the random graph G(n,p) obtained by allowing each edge in the complete graph on n vertices to be present with probability p independent of the other edges. In this paper, we study the minimum number of edge edit operations needed to convert G(n,p) into an Eulerian graph. We obtain deviation estimates for three types Eulerian edit numbers based on whether we perform only edge additions or only edge deletions or a combination of both and show that with high probability, roughly n4 operations suffice in all three cases.

The concept of spin ice graphs and a field theory for their charges

Originally detected in rare earth pyrochlores, spin ice physics is now being artificially extended to a variety of geometries that control collective behavior and exotic properties, making graph theory their proper framework. We relate spin ice notions, such as ice rule, ice manifold, Coulomb phases, charges, and monopoles, to graph-theoretical notions, such as balance, in/out-degrees, and Euler paths. We then propose a field-theoretical treatment in which topological charges and monopoles are the degrees of freedom, while the binary spins are subsumed in an entropic interaction among charges. We show that for a spin ice on a graph in a Gaussian approximation, the kernel of the entropic interaction is the inverse of the graph Laplacian, and we compute screening functions from the graph spectra as Green operators for the screened Poisson problem on a graph. We then apply the treatment to star graphs, tournaments, cycles, and regular spin ice in different dimensions. Our aim is twofold: to set spin ice physics in a proper graph setting, where only topological rather than geometrical notions hold, and to invite graph theorists to contribute their powerful tools to the field of spin ice.

A Study of Graph Theory Applications in IT Security

The recent developments in information technology have made major changes in all fields. The transfer of information through networks has become irreplaceable due to its advantages in facilitating the requirements of modern life through developing methods of storing and distributing information. This in turn has led to an increase in information problems and risks that threaten the security of the institution’s information and can be used in distributed systems environment.
 This study focused on two parts; the first is to review the most important applications of the graph theory in the field of network security, and the second is focused on the possibility of using the Euler graph as a Method Object that is employed in Remote Method Invocation (RMI) technique. This algorithm was compared with the most popular algorithms, such as RSA and 3DES. The results were acceptable and need to be improved in the future.

Almost eulerian compatible spanning circuits in edge-colored graphs

Let G be a (not necessarily properly) edge-colored graph. A compatible spanning circuit in G is a closed trail containing all vertices of G in which any two consecutively traversed edges have distinct colors. As two extremal cases, the existence of compatible (i.e., properly edge-colored) Hamilton cycles and compatible Euler tours have been studied extensively. More recently, sufficient conditions for the existence of compatible spanning circuits visiting each vertex v of G at least ⌊(d(v)−1)∕2⌋ times in graphs satisfying Ore-type degree conditions have been established. In this paper, we continue the research on sufficient conditions for the existence of compatible spanning circuits visiting each vertex at least a specified number of times. We respectively consider graphs satisfying Fan-type degree conditions, graphs with a high edge-connectivity, and the asymptotical existence of such compatible spanning circuits in random graphs.