Optimal parallel algorithm for the Hamiltonian cycle problem on dense graphs

G.A. Dirac´s classical theorem (1952) asserts that if every vertex of a graph G on n vertices has degree at least n /2, the G has a Hamiltonian cycle. A fast parallel algorithm on a concurrent-read-exclusive-write parallel random-access machine (CREW PRAM) is given to find a Hamiltonian cycle in such graphs. The algorithm uses a linear number of processors and is optimal up to a polylogarithmic factor. It works in O(log⁴n) parallel time and uses linear number of processors on a CREW PRAM. It is also proved that a perfect matching in dense graphs can be found in NC ². The cost of improved time is a quadratic number of processors. It is also proved that finding an NC algorithm for perfect matching in slightly less dense graphs is as hard as the same problem for all graphs, and the problem of finding a Hamiltonian cycle becomes NP-complete