Minimal Eulerian Circuit in a Labeled Digraph

2006 ◽  
pp. 737-744 ◽  
Author(s):  
Eduardo Moreno ◽  
Martín Matamala
Keyword(s):  
Eulerian Circuit

scholarly journals Rainbow eulerian multidigraphs and the product of cycles

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
Susana López ◽  
Francesc-Antoni Muntaner-Batle
Keyword(s):  
Eulerian Circuit ◽  
Vertex Set ◽  
International Audience

International audience An arc colored eulerian multidigraph with $l$ colors is rainbow eulerian if there is an eulerian circuit in which a sequence of $l$ colors repeats. The digraph product that refers the title was introduced by Figueroa-Centeno et al. as follows: let $D$ be a digraph and let $\Gamma$ be a family of digraphs such that $V(F)=V$ for every $F\in \Gamma$. Consider any function $h:E(D) \longrightarrow \Gamma$. Then the product $D \otimes_h \Gamma$ is the digraph with vertex set $V(D) \times V$ and $((a,x),(b,y)) \in E(D \otimes_h \Gamma)$ if and only if $(a,b) \in E(D)$ and $(x,y) \in E(h (a,b))$. In this paper we use rainbow eulerian multidigraphs and permutations as a way to characterize the $\otimes_h$-product of oriented cycles. We study the behavior of the $\otimes_h$-product when applied to digraphs with unicyclic components. The results obtained allow us to get edge-magic labelings of graphs formed by the union of unicyclic components and with different magic sums.

THE EULERIAN STRETCH OF A NETWORK TOPOLOGY AND THE ENDING GUARANTEE OF A CONVERGENCE ROUTING

2004 ◽  
Vol 05 (02) ◽  
pp. 93-109 ◽  
Author(s):  
DOMINIQUE BARTH ◽  
PASCAL BERTHOME ◽  
JOHANNE COHEN
Keyword(s):  
Optical Network ◽  
Combinatorial Design ◽  
Packet Routing ◽  
Lower And Upper Bounds ◽  
All Optical ◽  
Target Network ◽  
Eulerian Circuit ◽  
Number Formula ◽  
Network Topologies ◽  
Routing Techniques

In this paper, we focus on convergence packet routing techniques in an all-optical network, obtained from an Eulerian routing in the digraph modeling the target network. Given an Eulerian circuit [Formula: see text] in a digraph G, we deal with the maximal number [Formula: see text] of arcs that a packet has to follow on [Formula: see text] from its origin to its destination (we talk about the ending guarantee of the routing). We consider the Eulerian diameter of G as defined by [Formula: see text], where Eul(G) is the set of all the Eulerian circuits in G. After giving a preliminary result about the complexity of finding ℰ(G) for any digraph G, we give some lower and upper bounds of this parameter. The main part of the paper is devoted to the description of a combinatorial design of various network topologies having good Eulerian diameters.

scholarly journals Fuzzy Eulerian and Fuzzy Hamiltonian Graphs with Their Applications

2019 ◽  
Vol 8 (4) ◽  
pp. 7995-7999
Keyword(s):  
Set Theory ◽  
Hamiltonian Cycle ◽  
Real Life ◽  
Hamiltonian Graphs ◽  
Fuzzy Graphs ◽  
Eulerian Graphs ◽  
Eulerian Circuit ◽  
The Subject ◽  
Membership Value ◽  
Different Levels

In this article we discussed prominence of Fuzzy Eulerian and Fuzzy Hamiltonian graphs. Fuzzy logic is introduced to study the uncertainty of the event. In Fuzzy set theory we assign a membership value to each element of the set which ranges from 0 to 1. The earnest efforts of the researchers are perceivable in the relevant establishment of the subject integrating coherent practicality and reality. Fuzzy graphs found an escalating number of applications in day to day life system where the information intrinsic in the system varies with different levels of accuracy. In this article we initiated the model of fuzzy Euler graphs (FEG) and also fuzzy Hamiltonian graphs (FHG). We explored about fuzzy walk, fuzzy path, fuzzy bridge, fuzzy cut node, fuzzy tree, fuzzy blocks, fuzzy Eulerian circuit and fuzzy Hamiltonian cycle. Here we studied some applications of Fuzzy Eulerian graphs and fuzzy Hamiltonian graphs in real life.

You Can't Get There from Here—an Algorithmic Approach to Eulerian and Hamiltonian Circuits

1987 ◽  
Vol 80 (2) ◽  
pp. 95-148
Author(s):  
Joan H. Shyers
Keyword(s):  
Hamiltonian Circuit ◽  
Algorithmic Approach ◽  
Starting Point ◽  
Eulerian Circuit ◽  
Hamiltonian Circuits

Suppose you are in charge of the routing for a road crew whose job is to paint yellow lines down the center of all highways in a given section of the country. Your first task is to organize their travels to find a route that takes the crew over each section of the highway exactly once and returns them to their starting point. Such a route is an Eulerian circuit. Now suppose that the towns being served by these highways will pay the workers only on completion of the entire job. Your second task is to find a route that will take them through each town exactly once, again returning them to their starting point. This kind of route is a Hamiltonian circuit.

scholarly journals Graph Routing Problem Using Euler’s Theorem and Its Applications

2019 ◽  
Author(s):  
Hashnayne Ahmed
Keyword(s):  
Graph Theory ◽  
Social Science ◽  
Routing Protocols ◽  
Routing Problem ◽  
Euler’S Theorem ◽  
Bridge Problem ◽  
Eulerian Circuit ◽  
History Of ◽  
Proof Techniques ◽  
Modern Era

In this modern era, time and cases related to time is very important to us. For shortening time, Eulerian Circuit canopen a new dimension. In computer science, social science and natural science, graph theory is a stimulating space for thestudy of proof techniques. Graphs are also effective in modeling a variety of optimization cases like routing protocols, networkmanagement, stochastic approaches, street mapping etc. Konigsberg Bridge Problem has seven bridges linked with four islandsdetached by a river in such a way that one can’t walk through each of the bridges exactly once and returning back to thestarting point. Leonard Euler solved it in 1735 which is the foundation of modern graph theory. Euler’s solution forKonigsberg Bridge Problem is considered as the first theorem of Graph Theory which gives the idea of Eulerian circuit. It canbe used in several cases for shortening any path. From the Konigsberg Bridge Problem to ongoing DNA fragmentationproblem, it has its applications. Aiming to build such a dimension using Euler’s theorem and Konigsberg Bridge Problem, thispaper presents about the history of remarkable Konigsberg Bridge Problem, Euler’s Explanation on it, an alternativeexplanation and some applications to Eulerian Circuit using graph routing and Fortran Coding of it.

scholarly journals Graph Routing Problem Using Euler’s Theorem and Its Applications

2019 ◽  
Author(s):  
Hashnayne Ahmed
Keyword(s):  
Graph Theory ◽  
Social Science ◽  
Routing Protocols ◽  
Routing Problem ◽  
Euler’S Theorem ◽  
Bridge Problem ◽  
Eulerian Circuit ◽  
History Of ◽  
Proof Techniques ◽  
Modern Era

In this modern era, time and cases related to time is very important to us. For shortening time, Eulerian Circuit canopen a new dimension. In computer science, social science and natural science, graph theory is a stimulating space for thestudy of proof techniques. Graphs are also effective in modeling a variety of optimization cases like routing protocols, networkmanagement, stochastic approaches, street mapping etc. Konigsberg Bridge Problem has seven bridges linked with four islandsdetached by a river in such a way that one can’t walk through each of the bridges exactly once and returning back to thestarting point. Leonard Euler solved it in 1735 which is the foundation of modern graph theory. Euler’s solution forKonigsberg Bridge Problem is considered as the first theorem of Graph Theory which gives the idea of Eulerian circuit. It canbe used in several cases for shortening any path. From the Konigsberg Bridge Problem to ongoing DNA fragmentationproblem, it has its applications. Aiming to build such a dimension using Euler’s theorem and Konigsberg Bridge Problem, thispaper presents about the history of remarkable Konigsberg Bridge Problem, Euler’s Explanation on it, an alternativeexplanation and some applications to Eulerian Circuit using graph routing and Fortran Coding of it.

scholarly journals SOLUSI PERMASALAHAN KOMPUTASI SKALA BESAR PADA SOFTWARE ADOBE FLASH DALAM PERANCANGAN MEDIA PEMBELAJARAN PENCARIAN EULERIAN CIRCUIT

2016 ◽  
Vol 16 (1) ◽  
pp. 27-32
Author(s):  
Reza Wafdan ◽  
Mahyus Ihsan ◽  
Marwan Ramli ◽  
Hafnani Hafnani
Keyword(s):  
Graph Theory ◽  
Large Scale ◽  
Time Limit ◽  
Instructional Media ◽  
Adobe Flash ◽  
Eulerian Circuit ◽  
E Learning ◽  
Testing Algorithms ◽  
Programming Software ◽  
Better Than

Adobe Flash is a software that is used to build an interactive contents that can be attached in other things, such as presentations, games and e-learning. The Eulerian Circuit Search Instructional Media is one of examples in using Adobe Flash in building an interactive instructional media to solve graph theory problems. Visually, Adobe Flash is better than other programming software in general and it is the most suitable one for graph theory, as it uses graphics and visuals to solve problems. The weakness of Adobe Flash is when doing large scale calculations; such as when testing algorithms to find Eulerian circuits. Then an idea came about to complete the large scale calculations in Adobe Flash, which is commonly known as script time limit, with using other methodologies by changing script time limit on the settings, algorithms, partition the processes, and utulize the timers within the objects in the actionscript that handles the search algorithms of searching Eulerian circuit.

scholarly journals Realizations with a cut-through Eulerian circuit

1995 ◽  
Vol 137 (1-3) ◽  
pp. 265-275 ◽  
Author(s):  
Dal-Young Jeong
Keyword(s):  
Eulerian Circuit

scholarly journals Some graph properties of the optimised degree six 3-modified chordal ring network

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
Stephen Lim Een-Chien ◽  
R. N. Farah ◽  
M. Othman
Keyword(s):  
Optical Networks ◽  
Linear Array ◽  
Graph Model ◽  
Distributed Network ◽  
Circulant Graph ◽  
Ring Network ◽  
Hamiltonian Circuit ◽  
Graph Properties ◽  
Eulerian Circuit ◽  
Chordal Ring

The interconnection topology of a parallel or distributed network is pivotal in ensuring good system performance. It can be modelled by a graph, where its edges represent the links between processor nodes represented by vertices. One such graph model that has gained attention by researchers since its founding is the chordal ring, based on an undirected circulant graph. This paper discusses the degree six 3-modified chordal ring, CHR6o3, and presents its graph theoretical properties of symmetry and Hamiltonicity. CHR6o3 is shown to be asymmetric, and can be decomposed into similar subgraphs, each consisting of only one type of node in its class if ring links are ignored. These properties aid both the development of a routing scheme and also determining lower bounds for its chromatic number. Conditions for the existence of a Hamiltonian Circuit within CHR6o3 are also discussed. The existence of a Hamiltonian Circuit within a network simplifies parallel processing as the processors can be arranged to work on a task in a linear array. An Eulerian Circuit was shown to exist in CHR6o3. The existence of an Eulerian Circuit plays a role in routing in optical networks.

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