Show that none of graphs in 31-33 has a Hamiltonian circuit.
Show that the graph does not have a Hamiltonian circuit.
Given a graph G, from the subgraph consisting all vertices of degree 2 and their incident edges.
If G has a Hamiltonian circuit H (which we consider as a subgraph of G ), then it must contain as a subgraph, since a Hamiltonian circuit must include every edge incident on a vertex of degree 2.
If contains a vertex with degree > 2, then G has no Hamiltonian circuit.
If has a vertex of degree > 2, then so does H, which is impossible since H is a circuit.
If contains a circuit, but , then G has no Hamiltonian circuit
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