Posted by **Joy** on Tuesday, August 6, 2013 at 7:58pm.

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?

- Discrete Mathematics -
**MathMate**, Thursday, August 8, 2013 at 9:57pm
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.

- Discrete Mathematics -
**Joy**, Friday, August 9, 2013 at 1:36pm
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.

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