[Paper Review] Limited packings of closed neighbourhoods in graphs

Paul Balister, Béla Bollobás+1 more|arXiv (Cornell University)|Jan 8, 2015

Limits and Structures in Graph Theory6 references3 citations

TL;DR

This paper establishes tight lower bounds for the 2-limited packing number $L_2(G)$ in cubic graphs, proving $L_2(G) \geq |V(G)|/3$, which improves upon the prior $1/4$ bound. It further constructs extremal graphs using finite projective planes to show that existing asymptotic lower bounds are tight up to constants for fixed $k$ and growing maximum degree $\Delta$, and applies the Lovász Local Lemma to derive a nearly optimal bound when both $k$ and $\Delta$ grow rapidly.

ABSTRACT

The k-limited packing number, $L_k(G)$, of a graph $G$, introduced by Gallant, Gunther, Hartnell, and Rall, is the maximum cardinality of a set $X$ of vertices of $G$ such that every vertex of $G$ has at most $k$ elements of $X$ in its closed neighbourhood. The main aim in this paper is to prove the best-possible result that if $G$ is a cubic graph, then $L_2(G) \geq |V (G)|/3$, improving the previous lower bound given by Gallant, \emph{et al.} In addition, we construct an infinite family of graphs to show that lower bounds given by Gagarin and Zverovich are asymptotically best-possible, up to a constant factor, when $k$ is fixed and $Δ(G)$ tends to infinity. For $Δ(G)$ tending to infinity and $k$ tending to infinity sufficiently quickly, we give an asymptotically best-possible lower bound for $L_k(G)$, improving previous bounds.

Motivation & Objective

• To establish a best-possible lower bound for the 2-limited packing number $L_2(G)$ in cubic graphs.
• To analyze the asymptotic tightness of existing lower bounds for $L_k(G)$ when $k$ is fixed and $\Delta(G) \to \infty$.
• To improve asymptotic lower bounds for $L_k(G)$ when both $k$ and $\Delta(G)$ tend to infinity, using probabilistic techniques.
• To relate limited packing numbers to domination parameters in regular graphs, particularly via duality with $\ell$-tuple dominating sets.

Proposed method

• Prove that for any cubic graph $G$, $L_2(G) \geq |V(G)|/3$ using structural and extremal arguments, with equality achieved in a specific infinite family.
• Construct an infinite family of graphs based on finite projective planes to demonstrate that the Gagarin-Zverovich lower bound for $L_k(G)$ is tight up to a constant factor when $k$ is fixed and $\Delta \to \infty$.
• Apply the symmetric Lovász Local Lemma to derive a new asymptotic lower bound for $L_k(G)$ when $k > \log \Delta \log \log \Delta$ and $\Delta \to \infty$, showing $L_k(G) \geq \frac{k|V(G)|}{\Delta}(1+o(1))$.
• Use random sampling with Chernoff bounds to estimate the probability that a randomly chosen vertex set forms a $k$-limited packing, combining independence conditions and tail bounds.
• Leverage duality between $k$-limited packings and $(\Delta+1-k)$-tuple dominating sets in regular graphs to translate results into domination theory.
• Analyze the extremal case of $k$-limited packings in the graph $G_{q,k}$ formed from points and lines in projective geometry, showing $L_k(G_{q,k}) = k$ with $|V(G_{q,k})| \sim \frac{q^{k+1}}{q-1}$ and $\Delta(G_{q,k}) \sim \frac{q^k}{q-1}$, yielding $L_k(G_{q,k}) \sim \frac{k|V(G_{q,k})|}{\Delta(G_{q,k})^{1+1/k}}$.

Experimental results

Research questions

• RQ1 What is the best possible lower bound for $L_2(G)$ in cubic graphs, and is it tight?
• RQ2 How tight are existing asymptotic lower bounds for $L_k(G)$ when $k$ is fixed and $\Delta(G) \to \infty$?
• RQ3 Can improved asymptotic lower bounds for $L_k(G)$ be established when both $k$ and $\Delta(G)$ grow to infinity?
• RQ4 What is the relationship between $k$-limited packings and $\ell$-tuple dominating sets in regular graphs?
• RQ5 Are the known lower bounds for $L_k(G)$ asymptotically optimal, and if so, up to what constant factor?

Key findings

• For any cubic graph $G$, the 2-limited packing number satisfies $L_2(G) \geq |V(G)|/3$, and this bound is best-possible, as demonstrated by an explicit construction.
• The lower bound $L_k(G) \geq n \frac{k}{(k+1)\sqrt[k]{\binom{\Delta}{k}(\Delta+1)}}$ by Gagarin and Zverovich is asymptotically tight up to a constant factor when $k$ is fixed and $\Delta \to \infty$, as shown by constructing an infinite family of extremal graphs.
• When $k > \log \Delta \log \log \Delta$, the Lovász Local Lemma yields the asymptotically optimal bound $L_k(G) \geq \frac{k|V(G)|}{\Delta}(1+o(1))$ for graphs with maximum degree $\Delta$, which is tight up to lower-order terms.
• The bound $L_k(G) \geq \frac{k|V(G)|}{\Delta}(1+o(1))$ is asymptotically best possible for $\Delta$-regular graphs, as confirmed by the duality with $\ell$-tuple domination and the known upper bound $L_k(G) \leq \frac{k|V(G)|}{\delta(G)+1}$.
• In the graph $G_{q,k}$ derived from projective geometry, $L_k(G_{q,k}) = k$ and $|V(G_{q,k})| \sim \frac{q^{k+1}}{q-1}$, $\Delta(G_{q,k}) \sim \frac{q^k}{q-1}$, yielding $L_k(G_{q,k}) \sim \frac{k|V(G_{q,k})|}{\Delta(G_{q,k})^{1+1/k}}$, showing the bound $L_k(G) \geq \frac{k|V(G)|}{\Delta^{1+1/k}}$ is tight up to constants.
• The paper resolves the tightness of the $1/4$ bound from Gallant et al. for $L_2(G)$ in cubic graphs, replacing it with the optimal $1/3$ bound, and provides a new upper bound for 2-tuple dominating sets in cubic graphs via duality.