TY - JOUR

T1 - An extremal problem on degree sequences of graphs

AU - Linial, Nathan

AU - Rozenman, Eyal

PY - 2002

Y1 - 2002

N2 - Let G = (In, E) be the graph of the n-dimensional cube. Namely In = {0, 1}n and [x, y] ε E whenever ||x - y||I = 1. For A ⊆ In and x ε A define hA(x) = #{y ε In / A{[x, y] ε E}, 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 ∑xεA √hA(x) ≥ K for a universal constant K independent of n. We prove a related lower bound for graphs: Let G = (V, E) be a graph with |E| ≥ (k/2). Then ∑xεV(G) √d(x) ≥ k√k - 1, where d(x) is the degree of x. Equality occurs for the clique on k vertices.

AB - Let G = (In, E) be the graph of the n-dimensional cube. Namely In = {0, 1}n and [x, y] ε E whenever ||x - y||I = 1. For A ⊆ In and x ε A define hA(x) = #{y ε In / A{[x, y] ε E}, 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 ∑xεA √hA(x) ≥ K for a universal constant K independent of n. We prove a related lower bound for graphs: Let G = (V, E) be a graph with |E| ≥ (k/2). Then ∑xεV(G) √d(x) ≥ k√k - 1, where d(x) is the degree of x. Equality occurs for the clique on k vertices.

UR - http://www.scopus.com/inward/record.url?scp=0036972875&partnerID=8YFLogxK

U2 - 10.1007/s003730200041

DO - 10.1007/s003730200041

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:0036972875

SN - 0911-0119

VL - 18

SP - 573

EP - 582

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

IS - 3

ER -