A note on path kernels and partitions

The detour order of a graph G , denoted by τ ( 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 τ ( 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 τ ( G [ A ] ) ≤ a and τ ( 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 < 3 g 2 − 1 . Furthermore, if τ ( G ) = a + b , 1 ≤ a ≤ b , and G has girth greater than 2 3 ( a + 1 ) , then G has an ( a , b ) -partition.