Problems remaining NP-complette for sparse or dense graphs

For each flxed pair fi;c > 0 let INDEPENDENT SET (mcn fi ) and INDEPENDENT SET (m ‚ i n ¢ i cn fi ) be the problem INDE- PENDENT SET restricted to graphs on n vertices with mcn fi or m ‚ i n ¢ i cn fi edges, respectively. Analogously, HAMILTONIAN CIRCUIT (mn + cn fi ) and HAMILTONIAN PATH (mn + cn fi ) are the problems HAMILTONIAN CIRCUIT and HAMILTONIAN PATH restricted to graphs with mn+cn fi edges. For each † > 0 let HAMILTONIAN CIRCUIT (m ‚ (1i†) i n ¢ ) and HAMILTONIAN PATH (m ‚ (1 i †) i n ¢ ) be the problems HAMILTONIAN CIRCUIT and HAMILTONIAN PATH restricted to graphs with m ‚ (1i†) i n ¢ edges. We prove that these six restricted problems remain NP{complete. Finally, we consider su-cient conditions for a graph to have a Hamil- tonian circuit. These conditions are based on degree sums and neigh- borhood unions of independent vertices, respectively. Lowering the required bounds the problem HAMILTONIAN CIRCUIT jumps from 'easy' to 'NP{complete'.