Leonhard Paul Euler's Graph Theory

1.1 Introduction

Leonhard Paul Euler (1707-1783), a pioneering Swiss mathematician, who spent most of his life in Russia and Germany. Euler solved the first problem using graph theory and thereby led the foundation of very vast and important field of graph theory. He created first graph to simulate a real time place and situation to solve a problem which was then considered one of the toughest problems. That is, he solved the ‘Konigsberg bridge’ problem using graph theory and so the history of graph theory can be traced to 1736.

The ‘Konigsberg bridge’ problem originated in the city of Konigsberg, Prussia lies along the Pregel River. The river was divided the city of Konigsberg into four regions which are two large islands (A and D) and two

There are 6 vertices and 10 edges. An edge of a graph is said to be incident with a vertex if the vertex is an end vertex of the edge. Besides, two vertices of a graph are said to be adjacent if there is an edge joining them. For instance, the edge bd is incident with vertices b and d, while the vertices b and d are adjacent in the Figure 1.3.

A simple graph is a graph without multiple edges or self-loops. In Figure 1.4, it has shown a loop. A loop is an edge which joins a vertex itself and whose endpoints are equal. In Figure 1.5, it has shown multiple edges. Multiple edges are edges having the same pairs of endpoints.

The examples of simple graphs have shown in Figure 1.6. The cycle graph, path graph, complete graph, and wheel graphs all are the simple graphs. A cycle graph is denoted by Cn, which each vertex has degree 2. A path graph, Pn is a simple graph whose vertices can be ordered so that two vertices are adjacent if and only if they are consecutive in the list. A complete graph, Kn is a simple graph in which each pair of distinct vertices are adjacent. The graph obtained from Cn-1 by joining each vertex to a new vertex is the wheel on n vertices, denoted Wn. For a graph G, the number of vertices, v(G) is called the order of G, commonly denoted by |V(G)|, whereas the number of edges, e(G) is called the size of G, often denoted by