  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 $k \ge 2$ be a fixed integer,
$\alpha_1,...,\alpha_k$ be positive reals satisfying $\sum_{i} \alpha_i = 1$
and $(\alpha_1,..., \alpha_k) \neq (1/k,...,1/k)$, 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 $\alpha_1 n,..., \alpha_k n$, 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.
