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Saturated Subgraphs of the Hypercube

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  19 September 2016

J. ROBERT JOHNSON  and
TREVOR PINTO
J. ROBERT JOHNSON
Affiliation:
School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK (e-mail: r.johnson@qmul.ac.uk)
TREVOR PINTO
Affiliation:
School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK (e-mail: r.johnson@qmul.ac.uk)
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Abstract

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We say a graph is (Qn,Qm)-saturated if it is a maximal Qm-free subgraph of the n-dimensional hypercube Qn. A graph is said to be (Qn,Qm)-semi-saturated if it is a subgraph of Qn and adding any edge forms a new copy of Qm. The minimum number of edges a (Qn,Qm)-saturated graph (respectively (Qn,Qm)-semi-saturated graph) can have is denoted by sat(Qn,Qm) (respectively s-sat(Qn,Qm)). We prove that

$$ \begin{linenomath} \lim_{n\to\infty}\ffrac{\sat(Q_n,Q_m)}{e(Q_n)}=0, \end{linenomath}$$
for fixed m, disproving a conjecture of Santolupo that, when m=2, this limit is 1/4. Further, we show by a different method that sat(Qn, Q2)=O(2n), and that s-sat(Qn, Qm)=O(2n), for fixed m. We also prove the lower bound
$$ \begin{linenomath} \ssat(Q_n,Q_m)\geq \ffrac{m+1}{2}\cdot 2^n, \end{linenomath}$$
thus determining sat(Qn,Q2) to within a constant factor, and discuss some further questions.

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05D05: Extremal set theory
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 1 , January 2017 , pp. 52 - 67
Copyright
Copyright © Cambridge University Press 2016 

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