In 1960 Ghouila-Houri extended Dirac’s theorem to directed graphs by proving that if D is a directed graph on n vertices with minimum out-degree and in-degree at least n/2, then D contains a directed Hamiltonian cycle. For directed graphs one may ask for other orientations of a Hamiltonian cycle and in 1980 Grant initiated the problem of determining minimum degree conditions for a directed grap...

In 1997, Ng and Schultz introduced the idea of cycle orderability. For a positive integer k, a graph G is k-ordered if for every ordered sequence of k vertices, there is a cycle that encounters the vertices of the sequence in the given order. If the cycle is also a hamiltonian cycle, then G is said to be k-ordered hamiltonian. We give minimum degree conditions and sum of degree conditions for n...

In the Traveling Salesperson Problem, we are given an undirected graph G = (V,E) and cost c(e) > 0 for each edge e ∈ E. Our goal is to find a Hamiltonian cycle with minimum cost. A cycle is said to be Hamiltonian if it visits every vertex in V exactly once. TSP is known to be NP-complete, and so we cannot expect to exactly solve TSP in polynomial time. What is worse, there is no good approximat...

A 1-factorisation of a graph is perfect if the union of any two of its 1-factors is a Hamiltonian cycle. Let n=p for an odd prime p. We construct a family of (p−1)/2 non-isomorphic perfect 1-factorisations of Kn, n. Equivalently, we construct pan-Hamiltonian Latin squares of order n. A Latin square is pan-Hamiltonian if the permutation defined by any row relative to any other row is a single cy...

in this paper energy bands and berry curvature of graphene was studied. desired hamiltonian regarding the next-nearest neighbors obtained by tight binding model. by using the second quantization approach, the transformation matrix is calculated and the hamiltonian of system is diagonalized. with this hamiltonian, the band structure and wave function can be calculated. by using calculated wave f...