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A vertex coloring of a graphis called an  exact square coloring  of  G  if any pair of vertices at distance 2 receive distinct colors. The minimum number of colors required by any exact square coloring is called the  exact square chromatic number  of  G  and is denoted byA set of vertices is called an  exact square clique  of  G  if any pair of vertices of the set are at distance 2. The  exact square clique number  of  G , denoted byis the maximum cardinality of an exact square clique of  G  and clearly,In this article, we give tight upper bounds of the exact square chromatic number of various standard graph operations, including the Cartesian product, strong product, lexicographic product, corona product, edge corona product, join, subdivision, and Mycielskian of a graph. We also present tight lower bounds of the exact square clique number of these graph operations.

Exact square coloring of graphs resulting from some graph operations and products