The equivalence of two problems on the cube

C. Gotsman; N. Linial

doi:10.1016/0097-3165(92)90060-8

The equivalence of two problems on the cube

C. Gotsman^*, N. Linial

^*Corresponding author for this work

The Rachel and Selim Benin School of Engineering and Computer Science

Research output: Contribution to journal › Article › peer-review

20 Scopus citations

Abstract

Denote by Q_n the graph of the hypercube Cⁿ = { +1, -1}ⁿ. The following two seemingly unrelated questions are equivalent: 1. Let G be an induced subgraph of Q_n such that |V(G)| ≠ 2^n-1. Denote Δ(G) = max_{x ∈ V(G)}deg_G(x) and Γ(G) = max(Δ(G), Δ(Q_n - G)). Can Γ(G) be bounded from below by a function of n?; 2. Let f: Cⁿ → {+1, -1} be a boolean function. The sensitivity of f at x, denoted s(f, x), is the number of neighbors y of x in Q_n such that f(x) ≠ f(y). The sensitivity of f is s(f) = max_{x ∈ Cⁿ} s(f, x). Denote by d(f) the degree of the unique representation of f as a real multilinear polynomial on Cⁿ. Can d(f) be bounded from above by a function of s(f)?

Original language	English
Pages (from-to)	142-146
Number of pages	5
Journal	Journal of Combinatorial Theory. Series A
Volume	61
Issue number	1
DOIs	https://doi.org/10.1016/0097-3165(92)90060-8
State	Published - Sep 1992

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Funding Information:
* Supported in part by an Eshkol doctoral fellowship, Council for R&D, Israel Ministry of Science. + Research supported in part by US-Israel Binational Science Foundation and a grant from the Israel Academy of Sciences.

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title = "The equivalence of two problems on the cube",

abstract = "Denote by Qn the graph of the hypercube Cn = { +1, -1}n. The following two seemingly unrelated questions are equivalent: 1. Let G be an induced subgraph of Qn such that |V(G)| ≠ 2n-1. Denote Δ(G) = maxx ∈ V(G)degG(x) and Γ(G) = max(Δ(G), Δ(Qn - G)). Can Γ(G) be bounded from below by a function of n?; 2. Let f: Cn → {+1, -1} be a boolean function. The sensitivity of f at x, denoted s(f, x), is the number of neighbors y of x in Qn such that f(x) ≠ f(y). The sensitivity of f is s(f) = maxx ∈ Cn s(f, x). Denote by d(f) the degree of the unique representation of f as a real multilinear polynomial on Cn. Can d(f) be bounded from above by a function of s(f)?",

author = "C. Gotsman and N. Linial",

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N2 - Denote by Qn the graph of the hypercube Cn = { +1, -1}n. The following two seemingly unrelated questions are equivalent: 1. Let G be an induced subgraph of Qn such that |V(G)| ≠ 2n-1. Denote Δ(G) = maxx ∈ V(G)degG(x) and Γ(G) = max(Δ(G), Δ(Qn - G)). Can Γ(G) be bounded from below by a function of n?; 2. Let f: Cn → {+1, -1} be a boolean function. The sensitivity of f at x, denoted s(f, x), is the number of neighbors y of x in Qn such that f(x) ≠ f(y). The sensitivity of f is s(f) = maxx ∈ Cn s(f, x). Denote by d(f) the degree of the unique representation of f as a real multilinear polynomial on Cn. Can d(f) be bounded from above by a function of s(f)?

AB - Denote by Qn the graph of the hypercube Cn = { +1, -1}n. The following two seemingly unrelated questions are equivalent: 1. Let G be an induced subgraph of Qn such that |V(G)| ≠ 2n-1. Denote Δ(G) = maxx ∈ V(G)degG(x) and Γ(G) = max(Δ(G), Δ(Qn - G)). Can Γ(G) be bounded from below by a function of n?; 2. Let f: Cn → {+1, -1} be a boolean function. The sensitivity of f at x, denoted s(f, x), is the number of neighbors y of x in Qn such that f(x) ≠ f(y). The sensitivity of f is s(f) = maxx ∈ Cn s(f, x). Denote by d(f) the degree of the unique representation of f as a real multilinear polynomial on Cn. Can d(f) be bounded from above by a function of s(f)?

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The equivalence of two problems on the cube

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