In the well studied problem of Hamiltonicity, several papers/theorems gave sufficient degree conditions for the existence of Hamiltonian path in a graph.These include:Dirac's theorem , 1952, which states that A simple (undirected) graph with $n$ vertices ($n \geq 3$) is Hamiltonian if every vertex has degree $\frac{n}{2}$ or greater.Ghouila-Houiri 's theorem, (Une condition suffisante dexistence d'un circuit hamiltonien,1960), which generalized Dirac's result to directed graphs, showing that if $\delta^+(G),\delta^-(G) \geq \frac{n}{2}$ then G is Hamiltonian, where $\delta^+(G)$ is the minimal out degree in $G$, and by $\delta^-(G)$ the minimal in-degree.A theorem from Bang-Jensen and Gutin's book gave a stronger result, which imply that if $\forall v\in V: d_{in}(v)+d_{out}(v)\geq n$ then the graph is Hamiltonian.The $k$-path problem is a simple generalization of Hamiltonicity asking whether a simple path on $k$ nodes exist in a graph.What would be the weakest sufficient degree conditions from a graph such that it is bound to contain a $k$-path?
Is there a known extension of Dirac's / Ghoulia-Houri's theorems for $k$-path existence?
graph theory;hamiltonian paths