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Ex.3 (Petersen graph) See Figure 2. This is the famous Petersen graph which is a connected, undirected graph with 10 vertices and 15 edges. This exercise asks you to examine certain properties of the Petersen graph. A Hamiltonian path is a path in an undirected graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex exactly once. Answer the following with justification for the Petersen Graph: - Does it have a Hamiltonian path? - Does it have a Hamiltonian cycle? - Is it eulerian? - Is it bipartite? - Is it 2-colorable? Is it 3-colorable? A graph is 3-colorable if we can color every vertex with one of 3 colors (say, red, blue, green) so that no edge is monochromatic, i.e., every edge must have endpoints of different colors. - Show that if we delete any vertex from the Petersen graph, then the resulting graph will have a hamilton cycle. (hint: symmetry)