Quasi-randomness of graph balanced cut properties

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.