32. Which of the following is true of a Euler circuit? OIt cannot have any even vertices It can have at most 2 odd vertices O can have only one odd vertex it cannot have any odd vertices

a) A classmate claims that these two graphs are not isomorphic because Graph H has at least one simple circuit of length 6, but G does not have any simple circuit of length 6. Is this classmate's statement correct about the simple circuits? Answer yes or no and explain. Graph G Graph H b) Give two reasons of your own to explain why G and H are not isomorphic.

An Euler circuit of the graph K1 is, K1 (z, y, x, W, z, S, W) (V, z, X, W, z, s) (V, Z, x, W, z, s, W) a d. Not possible

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.

4. A graph G has 9 vertices and 21 edges. Three vertices have degree x, three have degree y, and three have degree z (where any of the x, y, and z are allowed to be the same number). a) How many graphs G could satisfy these criteria? b) How many graphs G could satisfy these criteria if G has no isolated vertices?