Problems remaining NP-complette for sparse or dense graphs

For each fixed pair α, c > 0 let INDEPENDENT SET (m ≤ cn) and INDEPENDENT SET (m ≥ n2 ) − cn) be the problem INDEPENDENT SET restricted to graphs on n vertices with m ≤ cn or m ≥ n2 )− cn edges, respectively. Analogously, HAMILTONIAN CIRCUIT (m ≤ n + cn) and HAMILTONIAN PATH (m ≤ n + cn) are the problems HAMILTONIAN CIRCUIT and HAMILTONIAN PATH restricted to graphs with m ≤ n + cn edges. For each 2 > 0 let HAMILTONIAN CIRCUIT (m ≥ (1− 2)n2 ) ) and HAMILTONIAN PATH (m ≥ (1 − 2)n2 ) ) be the problems HAMILTONIAN CIRCUIT and HAMILTONIAN PATH restricted to graphs with m ≥ (1− 2)n2 ) edges. We prove that these six restricted problems remain NP–complete. Finally, we consider sufficient conditions for a graph to have a Hamiltonian circuit. These conditions are based on degree sums and neighborhood unions of independent vertices, respectively. Lowering the required bounds the problem HAMILTONIAN CIRCUIT jumps from ’easy’ to ’NP–complete’.