This thesis puts forward the conjecture that for n > 3k with k > 2, the generalized Petersen graph, GP (n, k) is Hamilton-laceable if n is even and k is odd, and it is Hamilton-connected otherwise. We take the first step in the proof of this conjecture by proving the case n = 3k + 1 and k ≥ 1. We do this mainly by means of an induction which takes us from GP (3k + 1, k) to GP (3(k+2)+1, k+2). T...

Motivated by optimal control problems and differential games for functional equations of retarded type, the paper deals with a Cauchy problem path-dependent Hamilton--Jacobi equation right-end boundary condition. Minimax solutions this are studied. The existence uniqueness result is obtained under assumptions that weaker than those considered earlier. In contrast to previous works, on one hand,...

The symmetric group is generated by σ = (1 2 ··· n) and τ = (1 2). We answer an open problem of Nijenhuis and Wilf by constructing a Hamilton path in the directed Cayley graph for all n, and a Hamilton cycle for odd n. Dedicated to Herb Wilf (1931 – 2012).

An algorithm is described which constructs a long path containing a selected vertex x in a graph G. In hamiltonian graphs, it often finds a hamilton cycle or path. The algorithm uses crossovers of order k ≤ M , where M is a fixed constant, to build a longer and longer path. The method is based on theoretical methods often used to prove graphs hamiltonian.

A grid graph is a node-induced finite subgraph of the infinite grid. It is rectangular if its set of nodes is the product of two intervals. Given a rectangular grid graph and two of its nodes, we give necessary and sufficient conditions for the graph to have a Hamilton path between these two nodes. In contrast, the Hamilton path (and circuit) problem for general grid graphs is shown to be NP-co...

A bent Hamilton cycle in a grid graph is one in which each edge in a successive pair of edges lies in a different dimension. We show that the d-dimensional grid graph has a bent Hamilton cycle if some dimension is even and d ≥ 3, and does not have a bent Hamilton cycle if all dimensions are odd. In the latter case, we determine the conditions for when a bent Hamilton path exists. For the d-dime...

F. Ruskey Joe Sawada y May 15, 2002 Abstra t A bent Hamilton y le in a grid graph is one in whi h ea h edge in a su essive pair of edges lies in a di erent dimension. We show that the d-dimensional grid graph has a bent Hamilton y le if some dimension is even and d 3, and does not have a bent Hamilton y le if all dimensions are odd. In the latter ase, we determine the onditions for when a bent ...

This paper is devoted to the stochastic optimal control problem of ordinary differential equations allowing for both path-dependence and measurable randomness. As opposed deterministic path-dependent cases, value function turns out be a random field on path space it characterized by Hamilton–Jacobi (SPHJ) equation. A notion viscosity solution proposed proved unique associated SPHJ

We consider the minimum cycle factor problem: given a digraph D, find the minimum number kmin(D) of vertex disjoint cycles covering all vertices of D or verify that D has no cycle factor. There is an analogous problem for paths, known as the minimum path factor problem. Both problems are NP-hard for general digraphs as they include the Hamilton cycle and path problems, respectively. In 1994 Gut...