2243 The Ciência & Engenharia - Science & Engineering Journal ISSN: 0103-944X Volume 11 Issue 1, 2023 pp: 2243 - 2251 https://seer-ufu-br.online Special Graphs of Euler’s Family* and Tracing Algorithm- (A New Approach) Rajeshri Prajapati 1 , Amit Parikh 2 , Pradeep Jha 2* Department of Mathematics, Faculty of science, Ganpat University 1,2 Department of Mathematics, St. Xavier’s College, Gujrat University 2* rajeshriprajapati198@gmail.com 1 , amit.parikh.maths@gmail.com 2 , Corresponding author: Pradeep Jha 2* , pradeep.j.jha@gmail.com 2* Abstract: There are in graph theory, some known graphs which date back from centuries. [Euler graph, Hamiltonian graph etc.] These graphs are basic roots for development of graph theory. In this paper we have discussed the novel concept of tracing Euler tour. It depends on the concept of Link vertex - a join vertex of finite number of cycles as components of Euler graph. In addition to this, a new notion of isomorphic transformation of given graph on to a given line segment known as ‘Linear Graph’ also plays an important role for tracing the Euler grap h. Keywords: Odd edge, even edge, Link Vertex, Linear Graph, Line Graph, Mohmad scimitar, Devil’s Star Abbreviations: L(v), G(L) Assumption: The graph under discussion are undirected finite graphs Introduction: Graph theory was born in 1736 with Euler’s famous paper in which he solved the Konigsberg problem. Euler posed a more general problem in which for a given Euler graph G it is possible to find a closed walk running through every edge of G exactly once. A closed trail, containing all the edges is called an Euler trail. A closed connected graph having an Euler trail is called an Euler graph. Obviously in Euler graph, for every pair of points u and v there exist at least two edge disjoint u-v trails and consequently there are at least two edge –disjoint a u-v paths available to trace an Euler Path. There are some known algorithms for tracing like Fleury’s algorithm & Hierholzer’s algorithm. In this paper we have designed our own algorithm – J.P. algorithm, which works very efficiently. It is well known that Euler confronted with real life problem – seven bridges problem on the river Konigsberg, currently, a city in Russia. There were seven bridges on the river and it was referred to Euler that weather it is possible to traverse on all the seven bridges exactly once and reach the point of origin. Euler came out and successfully responding to the question. The very solution to this point was origin of the graph theory. The figure below shows the real physical situation and the corresponding graphical presentation of the problem on the river Pregel. Euler was confronted by commuters crossing different bridges to specific land areas of the banks and islands. The question was to suggest a path that initiates from a designated land area traverses on each bridge exactly once in one direction only and return the original point. [Figure 1: Konigsberg Bridge Problem]