Abstract.
Let G=(I n ,E) be the graph of the n-dimensional cube. Namely, I n ={0,1}n and [x,y]∈E whenever ||x−y||1=1. For A⊆I n and x∈A define h A (x) =#{y∈I n A|[x,y]∈E}, i.e., the number of vertices adjacent to x outside of A. Talagrand, following Margulis, proves that for every set A⊆I n of size 2n−1 we have for a universal constant K independent of n. We prove a related lower bound for graphs: Let G=(V,E) be a graph with . Then , where d(x) is the degree of x. Equality occurs for the clique on k vertices.
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Received: January 7, 2000
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ID="*" Supported in part by BSF and by the Israeli academy of sciences
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Linial, N., Rozenman, E. An Extremal Problem on Degree Sequences of Graphs. Graphs Comb 18, 573–582 (2002). https://doi.org/10.1007/s003730200041
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DOI: https://doi.org/10.1007/s003730200041