On the Vertex Identification Spectra of Grids

Abstract

Let [Formula: see text] be a red–white coloring of the vertices of a nontrivial connected graph [Formula: see text] with diameter [Formula: see text], where at least one vertex is colored red. Then [Formula: see text] is called an identification coloring or simply, an ID-coloring, if and only if for any two vertices [Formula: see text] and [Formula: see text], [Formula: see text], where for any vertex [Formula: see text], [Formula: see text] and [Formula: see text] is the number of red vertices at a distance [Formula: see text] from [Formula: see text]. A graph is said to be an ID-graph if it possesses an ID-coloring. If [Formula: see text] is an ID-graph, then the spectrum of [Formula: see text] is the set of all positive integers [Formula: see text] for which [Formula: see text] has an ID-coloring with [Formula: see text] red vertices. The identification number or ID-number of a graph is the smallest element in its spectrum. In this paper, we extend a result of Kono and Zhang on the identification number of grids [Formula: see text]. In particular, we give a formulation of strong ID-coloring and use it to give a sufficient condition for an ID-coloring of a graph to be extendable to an ID-coloring of the Cartesian product of a path [Formula: see text] with [Formula: see text]. Consequently, some elements of the spectrum of grids [Formula: see text] for positive integers [Formula: see text] and [Formula: see text], with [Formula: see text], are obtained. The complete spectrum of ladders [Formula: see text] is then determined using systematic constructions of ID-colorings of the ladders.