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

TL;DR

This paper determines the asymptotic domination number of a bipartite graph formed by 2-element and k-element subsets of an n-set, confirming a conjecture and extending previous results for the case of 1-element subsets.

Contribution

It proves a conjecture on the asymptotic domination number of the graph $G_{k,2}$, extending known results for $G_{k,1}$ and providing a precise asymptotic formula.

Findings

Asymptotic domination number of $G_{k,2}$ is ${k+3 ext{ over } 2(k-1)(k+1)} n^2 + o(n^2)

Confirmed the conjecture on the asymptotic value of the domination number for $G_{k,2}$

Extended previous results from $G_{k,1}$ to $G_{k,2}$

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 $\gamma (G_{k,2})={k+3\over 2(k-1)(k+1)}n^2+o(n^2)$.