[Paper Review] The domination number of the graph defined by two levels of the $n$-cube, II

József Balogh, Gyula O. H. Katona+2 more|arXiv (Cornell University)|Oct 23, 2019

Limits and Structures in Graph Theory4 citations

TL;DR

This paper proves a conjecture on the asymptotic domination number of the bipartite graph formed by the $k$-th and 2nd levels of the $n$-cube, establishing that $\gamma(G_{k,2}) = \frac{k+3}{2(k-1)(k+1)}n^2 + o(n^2)$ as $n \to \infty$ for fixed $k \geq 3$. The proof uses Frankl-R"odl hypergraph regularity and the Graph Removal Lemma to establish tight upper and lower bounds, resolving a long-standing open problem in extremal combinatorics.

ABSTRACT

Consider all $k$-element subsets and $\\ell$-element subsets $(k>\\ell )$ of an $n$-element set as vertices of a bipartite graph. Two vertices are adjacent if the corresponding $\\ell$-element set is a subset of the corresponding $k$-element set. Let $G_{k,\\ell}$ denote this graph. The domination number of $G_{k,1}$ was exactly determined by Badakhshian, Katona and Tuza. A conjecture was also stated there on the asymptotic value ($n$ tending to infinity) of the domination number of $G_{k,2}$. Here we prove the conjecture, determining the asymptotic value of the domination number $\\gamma (G_{k,2})={k+3\\over 2(k-1)(k+1)}n^2+o(n^2)$.

Motivation & Objective

• To resolve a conjecture on the asymptotic behavior of the domination number $\gamma(G_{k,2})$ for the bipartite graph formed by the $k$-th and 2nd levels of the $n$-cube.
• To establish tight asymptotic upper and lower bounds for $\gamma(G_{k,2})$ as $n \to \infty$ for fixed $k \geq 3$.
• To extend understanding of two-sided covering problems in extremal set theory, where both $k$-sets and $2$-sets must be dominated.
• To explore the potential tightness of lower bounds for higher $\ell$, particularly $\ell = 3$, and propose conjectures for $\gamma(G_{k,\ell})$.

Proposed method

• Uses Frankl-R"odl's hypergraph regularity theorem to construct a dominating set of size matching the conjectured asymptotic formula.
• Applies the Graph Removal Lemma to derive a lower bound on $\gamma(G_{k,2})$ by analyzing the density of pairs and their degrees in the hypergraph.
• Employs a partitioning strategy: divide $[n]$ into $k-1$ equal parts $A_1, \dots, A_{k-1}$, define $E$ as the set of pairs within each part, and use $F = \binom{[n]}{2} \setminus E$ as the domain for constructing $k$-sets.
• Constructs a family $\mathcal{K}$ of $k$-sets that intersect each $A_i$ in at least one element (with one exception having two), and defines $H(A)$ as the set of pairs from distinct $A_i$'s in $A$, ensuring coverage of all $k$-sets.
• Uses the degree and two-degree conditions in Theorem 2.1 to verify that the constructed hypergraph satisfies the requirements for covering $[N]$ with $\sim N/m$ sets.
• Provides two distinct proofs for the lower bound, one more detailed and insightful, both relying on the Graph Removal Lemma to show that no smaller dominating set can exist asymptotically.

Experimental results

Research questions

• RQ1 What is the asymptotic value of the domination number $\gamma(G_{k,2})$ as $n \to \infty$ for fixed $k \geq 3$?
• RQ2 Is the conjecture from Badakhshian, Katona, and Tuza on the asymptotic behavior of $\gamma(G_{k,2})$ correct?
• RQ3 Can the Graph Removal Lemma be effectively applied to derive tight lower bounds for domination numbers in bipartite graphs defined by two levels of the $n$-cube?
• RQ4 How do the structural properties of the $k$-sets and $2$-sets in the $n$-cube influence the minimal size of a dominating set?
• RQ5 Can similar methods be extended to determine $\gamma(G_{k,\ell})$ for $\ell > 2$, particularly for $\ell = 3$?

Key findings

• The paper proves that $\gamma(G_{k,2}) = \frac{k+3}{2(k-1)(k+1)}n^2 + o(n^2)$ as $n \to \infty$ for fixed $k \geq 3$, confirming the conjecture from prior work.
• The upper bound is established via a construction based on partitioning $[n]$ into $k-1$ equal parts and using hypergraph regularity theorems to ensure coverage of all $k$-sets and $2$-sets.
• The lower bound is derived using the Graph Removal Lemma, showing that any dominating set must have size at least $\frac{k+3}{2(k-1)(k+1)}n^2 + o(n^2)$, matching the upper bound.
• For $k=5, \ell=3$, the paper provides an upper bound of $\gamma(G_{5,3}) \leq \frac{1}{3}\binom{n}{3} + o(n^3)$, suggesting a potential asymptotic tightness.
• For $k=4, \ell=3$, the paper establishes $\gamma(G_{4,3}) \leq \frac{17}{27}\binom{n}{3} + o(n^3)$, with a construction based on 3-partitioning and degree analysis.
• The authors conjecture that the lower bound from Theorem 5.1 is asymptotically tight for all fixed $k > \ell \geq 2$ as $n \to \infty$, indicating a general pattern for $\gamma(G_{k,\ell})$.