Clumsy packings of graphs

Maria Axenovich, Anika Kaufmann Kaplan, Raphael Yuster

Research output: Contribution to journalArticlepeer-review


Let G and H be graphs. We say that P is an H-packing of G if P is a set of edge-disjoint copies of H in G. An H-packing P is maximal if there is no other H-packing of G that properly contains P. Packings of maximum cardinality have been studied intensively, with several recent breakthrough results. Here, we consider minimum cardinality maximal packings. An H-packing P is clumsy if it is maximal of minimum size. Let cl(G,H) be the size of a clumsy H-packing of G. We provide nontrivial bounds for cl(G,H), and in many cases asymptotically determine cl(G,H) for some generic classes of graphs G such as Kn (the complete graph), Qn (the cube graph), as well as square, triangular, and hexagonal grids. We asymptotically determine cl(Kn,H) for every fixed non-empty graph H. In particular, we prove that where ex(n,H) is the extremal number of H. A related natural parameter is cov(G,H), that is the smallest number of copies of H in G (not necessarily edge-disjoint) whose removal from G results in an H-free graph. While clearly cov(G,H) ≤ cl(G,H), all of our lower bounds for cl(G,H) apply to cov(G,H) as well.

Original languageEnglish
Article numberP2.39
JournalElectronic Journal of Combinatorics
Issue number2
StatePublished - 2019

Bibliographical note

Publisher Copyright:
© The authors.

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics
  • Applied Mathematics


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