On the strong path partition conjecture

Abstract

The detour order of a graph G, denoted by (G), is the order of a longest path in G. If a and b are positive integers and the vertex set of G can be partitioned into two subsets A and B such that (hAi) ≤ a and (hBi) ≤ b, we say that (A,B) is an (a, b)-partition of G. If equality holds in both instances, we call (A,B) an exact (a, b)-partition. The Path Partition Conjecture (PPC) asserts that if G is any graph and a, b any pair of positive integers such that (G) = a + b, then G has an (a, b)-partition. The Strong PPC asserts that under the same circumstances G has an exact (a, b)-partition. While a substantial body of work in support of the PPC has been developed over the past three decades, no results on the Strong PPC have yet appeared in the literature. In this paper we prove that the Strong PPC holds for a ≤ 8.