Final answer:
There can indeed be a bipartite graph with a circuit of length six, but there cannot be a simple graph with an odd number of vertices of odd degree due to the Handshaking Lemma. Thus, only statement a) is true.
Step-by-step explanation:
The question is asking whether two statements about graph theory are true. First, let's address a): there is a bipartite graph with a circuit of length six. This statement is true. A bipartite graph is a graph whose vertices can be divided into two disjoint sets such that every edge connects a vertex from one set to the other. Circuits in a bipartite graph must have even length, and six is an even number, so it is possible to have a circuit of length six in a bipartite graph.
Now we consider b): there is a simple graph with an odd number of vertices of odd degree. This statement is false due to the Handshaking Lemma, which states that in any graph, the sum of all the vertex degrees is twice the number of edges and therefore even. Thus, you cannot have an odd number of vertices of odd degree in a simple graph as it would imply an odd sum of degrees. The correct answer to the question is d) only a is true.