Question

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?

Answer 1

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.

Answer 2

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.