On Partitioning the Edge Set of a Graph into Internally Disjoint Paths without Exterior Vertices

Abstract

A collection of nontrivial paths in a graph G is called a path pile of G, if every edge of G is on exactly one path and no two paths have a common internal vertex. The least number that can be the cardinality of a path pile of G is called the path piling number of G. It can be shown that e ≤ ν + η where e, ν and η are respectively the size, the order and the path piling number of G. In this note we characterize structurally the class of all graphs for which the equality of this relation holds.