## Abstract

Denote by Q_{n} the graph of the hypercube C^{n} = { +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: C^{n} → {+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 C^{n}. Can d(f) be bounded from above by a function of s(f)?

Original language | American English |
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Pages (from-to) | 142-146 |

Number of pages | 5 |

Journal | Journal of Combinatorial Theory - Series A |

Volume | 61 |

Issue number | 1 |

DOIs | |

State | Published - Sep 1992 |

### Bibliographical note

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.