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

Abstract

AbstractA vertex coloring of a graph [Formula: see text] is 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 by [Formula: see text] A 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 by [Formula: see text] is the maximum cardinality of an exact square clique of G and clearly, [Formula: see text] 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.

Keywords

• Exact square coloring
• Cartesian product
• strong product
• corona product
• Mycielskian