Let G be a graph with the vertex set V = {v1, v2, v3, v4, v5}.

If the degrees of the vertices are 5, 1, 0, 6, 2, respectively, does G have an Eulerian path? Why or why not?

An Eulerian path is a path which passes through ALL edges exactly once.

This can happen if the graph has either 0 or 2 vertices with odd degrees.

So even an incomplete graph (with degree 0 at one or more vertices) can have an Eulerian path.

MathMate: Is this correct then for the following problem:

Let G be a graph with the vertex set V = [v1, v2, v3, v4, v5). If the degrees of the vertices are 5, 1, 0, 6, 2, respectively, does G have an Eulerian path? Why or why not?

5 + 1 + 0 + 6 + 2 = 14

2E = 14

E = 7

Yes, this problem does have an Eulerian path because it can have an odd degree or an incomplete graph can have the degree of 0 at one or more of the vertices.

To determine if graph G has an Eulerian path, we need to check the degrees of the vertices.

An Eulerian path is a path in a graph that traverses every edge exactly once, allowing for the possibility of revisiting vertices (except for the starting and ending vertices). For a graph to have an Eulerian path, it must satisfy certain conditions:

1. If the graph is connected, every vertex must have an even degree except for two vertices (the starting and ending vertices) that can have odd degrees.
2. If the graph is not connected, then every vertex except for the isolated vertices must have an even degree, and the isolated vertices can have any degree.

In this case, we have the degrees of the vertices as follows: 5, 1, 0, 6, 2.

Since three of the vertices have odd degrees (1, 5, and 1), condition 1 is not satisfied. Therefore, G does not have an Eulerian path.

To arrive at this conclusion, we calculated the degrees of the vertices. The degree of a vertex is the number of edges incident to it. By counting the number of edges connected to each vertex, we can determine the degree.