The Maximum 4-Vertex-Path Packing of a Cubic Graph Covers At Least Two-Thirds of Its Vertices

Abstract

Let \(P_4\) denote the path on four vertices. A \(P_4\) -packing of a graph G is a collection of vertex-disjoint copies of \(P_4\) in G. The maximum \(P_4\) -packing problem is to find a \(P_4\) -packing of maximum cardinality in a graph. In this paper, we prove that every simple cubic graph G on v(G) vertices has a \(P_4\) -packing covering at least \(\frac{2v(G)}{3}\) vertices of G and that this lower bound is sharp. Our proof provides a quadratic-time algorithm for finding such a \(P_4\) -packing of a simple cubic graph.