Graph Theory Questions Long
An Eulerian graph is a type of graph that contains a closed walk (a path that starts and ends at the same vertex) that traverses each edge exactly once. This closed walk is known as an Eulerian circuit or Eulerian cycle.
In simpler terms, an Eulerian graph is a graph in which we can start at any vertex, traverse each edge exactly once, and return to the starting vertex.
To determine if a graph is Eulerian, we need to check two conditions:
1. Connectedness: The graph must be connected, meaning that there is a path between any two vertices. If the graph is not connected, it can still be Eulerian if each connected component is Eulerian.
2. Degree of vertices: Every vertex in the graph must have an even degree. The degree of a vertex is the number of edges incident to it. If there are exactly two vertices with odd degrees, the graph can still be Eulerian, but it will not have an Eulerian circuit. Instead, it will have an Eulerian path, which starts at one of the odd-degree vertices and ends at the other.
Eulerian graphs have several interesting properties and applications. For example, they can be used to solve the famous Seven Bridges of Königsberg problem, which led to the development of graph theory. Eulerian circuits also have applications in network routing, DNA sequencing, and scheduling problems.