Vertex Identification in Grids and Prisms

Abstract

A red-white coloring of a nontrivial connected graph [Formula: see text] of diameter [Formula: see text] is an assignment of red and white colors to the vertices of [Formula: see text] where at least one vertex is colored red. Associated with each vertex [Formula: see text] of [Formula: see text] is a [Formula: see text]-vector, called the code of [Formula: see text], whose [Formula: see text]th coordinate is the number of red vertices at distance [Formula: see text] from [Formula: see text]. A red-white coloring of [Formula: see text] for which distinct vertices have distinct codes is called an identification coloring or ID-coloring of [Formula: see text]. A graph [Formula: see text] possessing an ID-coloring is an ID-graph. The minimum number of red vertices among all ID-colorings of an ID-graph [Formula: see text] is the identification number or ID-number of [Formula: see text]. It is shown that the grid [Formula: see text] is an ID-graph if and only [Formula: see text] and the prism [Formula: see text] is an ID-graph if and only if [Formula: see text].