Distributing vertices on Hamiltonian cycles

On the mutually independent Hamiltonian cycles in faulty hypercubes

Two ordered Hamiltonian paths in the n-dimensional hypercube Qn are said to be independent if i-th vertices of the paths are distinct for every 1 ≤ i ≤ 2n. Similarly, two s-starting Hamiltonian cycles are independent if i-th vertices of the cycle are distinct for every 2 ≤ i ≤ 2n. A set S of Hamiltonian paths and sstarting Hamiltonian cycles are mutually independent if every two paths or cycles...

Distributing vertices along a Hamiltonian cycle in Dirac graphs

Counting Hamiltonian cycles on planar random lattices

Hamiltonian cycles on planar random lattices are considered. The generating function for the number of Hamiltonian cycles is obtained and its singularity is studied. Hamiltonian cycles have often been used to model collapsed polymer globules[1]. A Hamiltonian cycle of a graph is a closed path which visits each of the vertices once and only once. The number of Hamiltonian cycles on a lattice cor...

Random partitions and edge-disjoint Hamiltonian cycles

Nash-Williams [21] proved that every graph with n vertices and minimum degree n/2 has at least b5n/224c edge-disjoint Hamiltonian cycles. In [20], he raised the question of determining the maximum number of edge-disjoint Hamiltonian cycles, showing an upper bound of b(n+ 4)/8c. Let α(δ, n) = (δ + √ 2δn− n2)/2. Christofides, Kühn, and Osthus [7] proved that for every > 0, every graph G on a suff...

Perfect matchings and Hamiltonian cycles in the preferential attachment model

We study the existence of perfect matchings and Hamiltonian cycles in the preferential attachment model. In this model, vertices are added to the graph one by one, and each time a new vertex is created it establishes a connection with m random vertices selected with probabilities proportional to their current degrees. (Constant m is the only parameter of the model.) We prove that if m ≥ 1,260, ...