If a graph contains an Euler circuit, what must be true of the degrees of the vertices of that graph?

13. Consider the following graph: B F E Does it have an Euler circuit, an Euler path, or neither? Explain why.

Determine an Euler circuit that begins with an vertex C in this Graph. 

Explain why the graph does or does not have an Euler circuit. If it does, specify the nodes within the circuit Figure 9: An undirected graph has 7 vertices, a through g. 5 vertices are in the form of a regular pentagon, rotated 90 degrees clockwise. Hence, the top vertex becomes the rightmost vertex. From the bottom left vertex, moving clockwise, the vertices in the pentagon shape are labeled: a, b, c, e, and f. Vertex d is above vertex e, below and to the right of vertex c. Vertex g is below vertex e, above and to the right of vertex f. Undirected edges, line segments, are between the following vertices: a and b; a and c; a and f; b and f; b and c; c and d; c and g; d and e; d and f; and f and g.

By duplicating only 2 edges, Eulerize the following graph:

Is it possible for a graph to have both a Euler path and Euler circuit? Explain your answer.

Prove that if there is a circuit in a graph that starts and ends at a vertex v and if w is another vertex in the circuit, then there is a circuit in the graph that starts and ends at w.