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 -