Let denote the number of vertices in a longest path of the graph . A subset of is called a -\emph{kernel} of if and every vertex is adjacent to an end-vertex of a path of order in .
A partition of is called an -partition if and .
We show that any graph with girth greater than has a -kernel and that every graph has a -kernel. As corollaries of these results, we show that if and has girth greater than or , then has an -partition.