TY - JOUR
T1 - Quasi-randomness of graph balanced cut properties
AU - Huang, Hao
AU - Lee, Choongbum
PY - 2012/8
Y1 - 2012/8
N2 - Quasi-random graphs can be informally described as graphs whose edge distribution closely resembles that of a truly random graph of the same edge density. Recently, Shapira and Yuster proved the following result on quasi-randomness of graphs. Let k ≥ 2 be a fixed integer, α 1,...,α k be positive reals satisfying ∑ iα i = 1 and (α 1,...,α k)≠(1/k,...,1/k), and G be a graph on n vertices. If for every partition of the vertices of G into sets V 1,...,V k of size α 1n,...,α kn, the number of complete graphs on k vertices which have exactly one vertex in each of these sets is similar to what we would expect in a random graph, then the graph is quasi-random. However, the method of quasi-random hypergraphs they used did not provide enough information to resolve the case (1/k,...,1/k) for graphs. In their work, Shapira and Yuster asked whether this case also forces the graph to be quasi-random. Janson also posed the same question in his study of quasi-randomness under the framework of graph limits. In this paper, we positively answer their question.
AB - Quasi-random graphs can be informally described as graphs whose edge distribution closely resembles that of a truly random graph of the same edge density. Recently, Shapira and Yuster proved the following result on quasi-randomness of graphs. Let k ≥ 2 be a fixed integer, α 1,...,α k be positive reals satisfying ∑ iα i = 1 and (α 1,...,α k)≠(1/k,...,1/k), and G be a graph on n vertices. If for every partition of the vertices of G into sets V 1,...,V k of size α 1n,...,α kn, the number of complete graphs on k vertices which have exactly one vertex in each of these sets is similar to what we would expect in a random graph, then the graph is quasi-random. However, the method of quasi-random hypergraphs they used did not provide enough information to resolve the case (1/k,...,1/k) for graphs. In their work, Shapira and Yuster asked whether this case also forces the graph to be quasi-random. Janson also posed the same question in his study of quasi-randomness under the framework of graph limits. In this paper, we positively answer their question.
KW - Cut properties
KW - Pseudo-random graphs
KW - Quasi-random graphs
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U2 - 10.1002/rsa.20384
DO - 10.1002/rsa.20384
M3 - Article
AN - SCOPUS:84862182714
SN - 1042-9832
VL - 41
SP - 124
EP - 145
JO - Random Structures and Algorithms
JF - Random Structures and Algorithms
IS - 1
ER -