Various variants of Hamiltonian problems

Hamiltonian cycle and Hamiltonian path problems are famous hard problems. The Hamiltonian cycle seems to have received more attention from the mathematics community. In this short note, we want to mitigate this bias a little bit. Keeping on track with the Prasolov and Sharygin kinds of doing mathematics, we give several simple constructions to show the hardness of some variants of Hamiltonian path problems.

Winding indexes of Max. and Min. Hamiltonians in N-Gons

The resolutions of the different Shortest and Longest Euclidean Hamiltonian Path Problems on the vertices of simple regular [Formula: see text]-Gons, by means of a geometric and arithmetic algorithm allow us to define winding indexes for Euclidean Hamiltonian cycles. New statements characterize orientation of non-necessarily regular Hamiltonian cycles on the [Formula: see text]th roots of the unity embedded in the plane and deal with the existence of reflective bistarred Hamiltonian tours on vertices of coupled [Formula: see text]-Gons.

EVERY LONGEST HAMILTONIAN PATH IN EVEN n-GONS

We single out every longest path of n-1 order that solves each of the [Formula: see text] Longest Euclidean Hamiltonian Path Problems on the even nth root of the unity, by means of a geometric and arithmetic procedure. This identification is done regardless of planar rotations and orientation. In addition, the uniqueness of the Euclidean Hamiltonian cycle that resolves the Maximum Traveling Salesman Problem is shown.