The Eulerian quandry

Euler path is one of the most discussed topic in Graph traversals. While finding Euler path in a graph, one may halt unsucessfully at a vertex with some of the vertices (and edges also) unvisited. To avoid this unlikely situation we have proposed some precautionary steps that one should have taken in account to explore a smooth Euler path without being halt at any vertex. Examples are given with complete demonstration of our proposed approach.

2023

This study introduces a groundbreaking concept in graph theory: Edge Connectivity, which aids in predicting the Eulerity of a line graph without the need to draw the line graph ()of a connected graph G. The paper not only delineates this novel concept but also discusses its salient features and applications, emphasizing its importance in graph theory. Through rigorous theoretical analysis, the study demonstrates the potential of Edge Connectivity in simplifying and enhancing the understanding of complex graph relationships, thereby opening new avenues for research in this field.

A straight-ahead walk in an embedded Eulerian graph G always passes from an edge to the opposite edge in the rotation at the same vertex. A straight-ahead walk is called Eulerian if all the edges of the embedded graph G are traversed in this way starting from an arbitrary edge. An embedding that contains an Eulerian straight-ahead walk is called an Eulerian embedding. In this article, we characterize some properties of Eulerian embeddings of graphs and of embeddings of graphs such that the corresponding medial graph is Eulerian embedded. We prove that in the case of 4-valent planar graphs, the number of straight-ahead walks does not depend on the actual embedding in the plane. Finally, we show that the minimal genus over Eulerian embeddings of a graph can be quite close to the minimal genus over all embeddings.

Abstract In this paper we define the partition of graph and introduce some of Eulerian graph properties through Adjacency and Degree matrices. We prove that each Eulerian graph such as G = (n, m) can be divided into s = (n - 1)/2 isomorphic Eulerian subgraphs and we conclude that the sufficient condition for a graph to be non-Eulerian is that the determinant of its adjacency matrix is an odd integer. We also put an effective algorithm to find all edge-disjoint cycles in any Eulerian graph.

Main objective of this paper to study Euler graph and it's various aspects in our real world. Now a day's Euler graph got height of achievement in many situations that occur in computer science, physical science, communication science, economics and many other areas can be analysed by using techniques found in a relatively new area of mathematics. The graphs concerns relationship with lines and points (nodes). The Euler graph can be used to represent almost any problem involving discrete arrangements of objects where concern is not with the internal properties of these objects but with relationship among them. To achieve objective I first study basic concepts of graph theory, after that I summarizes the methods that are adopted to find Euler path and Euler cycle.

Evgeny Zaytsev 1 PREPRINT 1 A short paper by Leonhard Euler "The solution of a problem belonging to the geometria situs, written in 1736, is usually referred to as the first substantial publication in topology and graph theory. 2 Expressing a common opinion, Emil A. Fellmann writes, for example:

International Journal of Applied Science and …, 2012

This article reflects some theoretical results related with Euler Graph obtained from a special pattern of Euler Diagrams.

Engineering Mathematics, 2019

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

Lecture Notes in Computer Science, 2004

Drawing Graphs in Euler Diagrams Paul Mutton1, Peter Rodgers1, and Jean Flower2 ... We use several criteria for measuring diagram features, such as contour smoothness, contour size, zone area and contour closeness. The criteria and the hill climber are described in [8]. Fig. ...