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Saturated Subgraphs of the Hypercube

Part of: Graph theory Extremal combinatorics

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

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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References

[1] Alon, N., Krech, A. and Szabó, T. (2007) Turán's theorem in the hypercube. SIAM J. Discrete Math. 21 66–72.Google Scholar
[2] Balogh, J., Hu, P., Lidický, B. and Liu, H. (2014) Upper bounds on the size of 4- and 6-cycle-free subgraphs of the hypercube. European J. Combin. 35 75–85.CrossRefGoogle Scholar
[3] Brass, P., Harborth, H. and Nienborg, H. (1995) On the maximum number of edges in a C 4-free subgraph of Q n . J. Graph Theory 19 17–23.Google Scholar
[4] Choi, S. and Guan, P. (2008) Minimum critical squarefree subgraph of a hypercube. Congressus Numerantium 189 57–64.Google Scholar
[5] Erdős, P. (1984) Some problems in graph theory, combinatorial analysis and combinatorial number theory. In Graph Theory and Combinatorics (Bollobás, B., ed.), Academic Press, pp. 1–17.Google Scholar
[6] Erdős, P., Hajnal, A. and Moon, J. W. (1964) A problem in graph theory. Amer. Math. Monthly 71 1107–1110.Google Scholar
[7] Faudree, J. R., Faudree, R. J. and Schmitt, R. (2011) A survey of minimum saturated graphs. Electron. J. Combin. 18.Google Scholar
[8] Katona, G. O. H. and Tarján, T. G. (1983) Extremal problems with excluded subgraphs in the n-cube. In Graph Theory: Łagów, Poland, Vol. 1018 of Lecture Notes in Mathematics, Springer, pp. 84–93.Google Scholar
[9] van Lint, J. H. (1999) Introduction to Coding Theory, Springer.Google Scholar
[10] Morrison, N., Noel, J. A. and Scott, A. (2014) On saturated k-Sperner systems. Electron. J. Combin. 21, Paper #P3.22.CrossRefGoogle Scholar
[11] Pikhurko, O. (2004) Results and open problems on minimum saturated hypergraphs. Ars Combinatorica 72 435–451.Google Scholar