Clumsy packings of graphs

TL;DR

This paper investigates minimal maximal packings of graphs with subgraphs isomorphic to a fixed graph, providing bounds and asymptotic formulas for various classes including complete graphs and grids, extending understanding of graph packing parameters.

Contribution

It introduces the concept of clumsy packings, derives bounds, and asymptotically determines their size for several important graph classes, linking to extremal graph theory.

Findings

Derived bounds for clumsy packings in various graphs

Asymptotic formulas for clumsy packings in complete graphs

Connection between clumsy packings and extremal numbers

Abstract

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 $K_n$ (the complete graph), $Q_n$ (the cube graph), as well as square, triangular, and hexagonal grids. We asymptotically determine $cl(K_n,H)$ for every fixed non-empty graph $H$. In particular, we prove that $$ cl(K_n, H) = \frac{\binom{n}{2}- ex(n,H)}{|E(H)|}+o(ex(n,H)),$$ 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 $cov(G,H) \le cl(G,H)$, all of our lower bounds for $cl(G,H)$ apply to $cov(G,H)$ as well.