3. Paths and cycles in a graph A path P -(. ) of length k is an (vi, Vi+) exists for 0iSk-1 and all the edges are different. A cycle C-(w.vi, , th-i,%) of length k is a path that starts and ends with the sane vertex In a simple path all the vertices are different. In a simple cycle all the vertices except k are different. An Euler path is a path that contains all the edges in the graph. An Euler cycle is a cycle that contains all the edges A Hamiltonian path is a simple path that contains all the vertices in the graph. A Hamiltonian cycle is a simple cycle that contains all the vertices in the graph. The Petersen graph does not have neither an Euler path nor an Euler cycle It has a Hamiltonian path but not a Hamiltonian cycle. (a) Find one of the longest paths (does not have to be simple) in the Petersen graph. (b) Find one of the longest cycles (does not have to be simple) in the Petersen graph. (c) Find one of the Hamiltonian paths in the Petersen graph. (d) Find one of the longest simple cycles in the Petersen graph. 3. Paths and cycles in a graph A path P -(. ) of length k is an (vi, Vi+) exists for 0iSk-1 and all the edges are different. A cycle C-(w.vi, , th-i,%) of length k is a path that starts and ends with the sane vertex In a simple path all the vertices are different. In a simple cycle all the vertices except k are different. An Euler path is a path that contains all the edges in the graph. An Euler cycle is a cycle that contains all the edges A Hamiltonian path is a simple path that contains all the vertices in the graph. A Hamiltonian cycle is a simple cycle that contains all the vertices in the graph. The Petersen graph does not have neither an Euler path nor an Euler cycle It has a Hamiltonian path but not a Hamiltonian cycle. (a) Find one of the longest paths (does not have to be simple) in the Petersen graph. (b) Find one of the longest cycles (does not have to be simple) in the Petersen graph. (c) Find one of the Hamiltonian paths in the Petersen graph. (d) Find one of the longest simple cycles in the Petersen graph