On strict-double-bound graphs and Cartesian products of paths and cycles

Abstract

For a poset [Formula: see text] the strict-double-bound graph ([Formula: see text]) of [Formula: see text] is the graph with the vertex set [Formula: see text] such that [Formula: see text] if and only if [Formula: see text] and there exist [Formula: see text] and [Formula: see text] distinct from [Formula: see text] and [Formula: see text] such that [Formula: see text] and [Formula: see text] For a connected graph [Formula: see text], the strict-double-bound number [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the graph with [Formula: see text] vertices and no edges. In this paper we deal with the strict-double-bound numbers of Cartesian products of graphs. We show that [Formula: see text] for [Formula: see text], [Formula: see text] for [Formula: see text], and [Formula: see text] for [Formula: see text].