A fast parallel algorithm for finding Hamiltonian cycles in dense graphs

Suppose that 0 < η < 1 is given. We call a graph, G , on n vertices an η -Chvátal graph if its degree sequence d 1 ≤ d 2 ≤ ⋯ ≤ d n satisfies: for k < n / 2 , d k ≤ min { k + η n , n / 2 } implies d n − k − η n ≥ n − k . (Thus for η = 0 we get the well-known Chvátal graphs.) An NC 4 -algorithm is presented which accepts as input an η -Chvátal graph and produces a Hamiltonian cycle in G as an output. This is a significant improvement on the previous best NC -algorithm for the problem, which finds a Hamiltonian cycle only in Dirac graphs ( δ ( G ) ≥ n / 2 where δ ( G ) is the minimum degree in G ).