Asymptotically optimal induced decompositions

Abstract

Solving a problem raised by Bondy and Szwarcfiter [J. Graph Theory, 72 (2013), 462-477] we prove that if the edge set of a graph G of order n can be decomposed into edge-disjoint induced copies of the path P4 or of the paw K4?P3, then the complement of G has at least cn3/2 edges. This lower bound is tight apart from the actual value of c, and completes the determination of asymptotic growth for the graphs with at most four vertices. More generally the lower bound cn3/2 holds for any graph without isolated vertices which is not a complete multipartite graph; but a linear upper bound is valid for any complete tripartite graph.