Show that all vertices of an Euler graph G are of even degree

To show that all vertices of an Euler graph G are of even degree, we can use the following steps:

Step 1: Recall that an Euler graph is a graph that contains a closed trail (a trail that starts and ends at the same vertex) that includes every edge exactly once.

Step 2: Let's assume that there exists a vertex in G that has an odd degree. This means that there is an odd number of edges incident to that vertex.

Step 3: Since every edge is counted twice when calculating the sum of degrees of all vertices in a graph, the sum of degrees of all vertices in G would be odd.

Step 4: However, the sum of degrees of all vertices in a graph must always be even. This is because each edge contributes exactly two to the sum of degrees (one for each of its incident vertices).

Step 5: Therefore, our assumption that there exists a vertex in G with an odd degree leads to a contradiction.

Step 6: Hence, we can conclude that all vertices of an Euler graph G are of even degree.

By following these steps, we have shown that all vertices of an Euler graph are of even degree.