An Extremal Problem on Degree Sequences of Graphs

Let G ðIn; EÞ be the graph of the n-dimensional cube. Namely, In f0; 1g n and x; yŠ 2E whenever jjx yjj1 1. For A In and x 2 A define hAðxÞ #fy 2 In n Ajx; yŠ 2Eg, i.e., the number of vertices adjacent to x outside of A. Talagrand, following Margulis, proves that for every set A In of size 2n 1 we have 1 2n P pffiffiffiffiffiffiffiffiffiffiffi x2A hAðxÞ K for a universal constant K independent of n. We prove a related lower bound for graphs: Let k P pffiffiffiffiffiffiffiffiffi G ðV; EÞ be a graph with jEj 2. Then x2V ðGÞ dðxÞ k ffiffiffiffiffiffiffiffiffiffiffi p k 1, where dðxÞ is the degree of x. Equality occurs for the clique on k vertices