We consider sufficient conditions for the existence of $k$-th powers of Hamiltonian cycles in $n$-vertex graphs $G$ with minimum degree $\mu n$ for arbitrarily small $\mu>0$. About 20 years ago Koml\'os, Sark\"ozy, and Szemer\'edi resolved the conjectures of P\'osa and Seymour and obtained optimal minimum degree conditions for this problem by showing that $\mu=\frac{k}{k+1}$ suffices for large $n$... (read more)
PDF