Bent Hamilton cycles in \(d\)-dimensional grid graphs (Q1856346)

A bent Hamilton cycle in a grid graph is one on which each edge in a successive pair of edges lies in a different dimension. In the paper it is shown that the \(d\)-dimensional grid graph has a bent Hamilton cycle if some dimension is even and \(d\geq 3\), and it does not have a bent Hamilton cycle if all dimensions are odd. In the latter case, conditions are determined for a bent Hamilton path to exist. For the \(d\)-dimensional toroidal grid graph (i.e., the graph product of \(d\) cycles), the authors show that there exists a bent Hamilton cycle when all dimensions are odd and \(d\geq 3\). It is also shown that if \(d=2\), then there exists a bent Hamilton cycle if and only if both dimensions are even.