Asymptotically Effective Method to Explore Euler Path in a Graph

Euler path is one of the most interesting and widely discussed topics in graph theory. An Euler path (or Euler trail) is a path that visits every edge of a graph exactly once. Similarly, an Euler circuit (or Euler cycle) is an Euler trail that starts and ends on the same node of a graph. A graph having Euler path is called Euler graph. While tracing Euler graph, one may halt at arbitrary nodes while some of its edges left unvisited. In this paper, we have proposed some precautionary steps that should be considered in exploring a deadlock-free Euler path, i.e., without being halted at any node. Simulation results show that our proposed approach improves the process of exploring the Euler path in an undirected connected graph without interruption. Furthermore, our proposed algorithm is complete for all types of undirected Eulerian graphs. The paper concludes with the proofs of the correctness of proposed algorithm and its computation complexity.

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.

Study on Necessary and Sufficient Conditions for Euler Graph and Hamilton Graph

Three theorems are proposed in this paper. The first theorem is that a connected undirected graph G is an Euler graph if and only if G can be expressed as the union of two circles without overlapped sides. Namely, equation satisfies. The second theorem is that a connected simple undirected graph is a Hamilton graph if and only if G contains a sub-graph generated by union of circles of sub-graphs derived from two endpoints of common side. Namely, the equation satisfies (meaning of symbols in the equations see main body of this paper). The third theorem is that a connected simple undirected graph is a Hamilton graph if and only if the loop sum of two circles, and, of sub-graphs derived from two endpoints of common side in graph G is a sub-graphs of loop graph Cn.

A generalization of the Königsberg bridges problem

A multi-edge graph is said to be a restricted Euler graph when it can be decomposed into cycles of length greater than 2. Necessary and sufficient conditions are obtained in relation to vertex structure for a graph to be restricted Euler.