A note on path kernels and partitions

The detour order of a graph G, denoted by @t(G), is the order of a longest path in G. A subset S of V(G) is called a P"n-kernel of G if @t(G[S])@?n-1 and every vertex v@?V(G)-S is adjacent to an end-vertex of a path of order n-1 in G[S]. A partition of the vertex set of G into two sets, A and B, such that @t(G[A])@?a and @t(G[B])@?b is called an (a,b)-partition of G. In this paper we show that any graph with girth g has a P"n"+"1-kernel for every n<3g2-1. Furthermore, if @t(G)=a+b, 1@?a@?b, and G has girth greater than 23(a+1), then G has an (a,b)-partition.