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

For a poset P = (X , ≤ P) , the strict-double-bound graph (sDB (P)) of P = (X , ≤ P) is the graph with the vertex set X such that u v ∈ E (sDB (P)) if and only if u ≠ v and there exist x ∈ X and y ∈ X distinct from u and v such that x ≤ P u ≤ P y and x ≤ P v ≤ P y. For a connected graph G , the strict-double-bound number ζ (G) is defined as min { l : there exists a poset P such that sDB(P) ≅ G ∪ N l } , where N l is the graph with l vertices and no edges. In this paper we deal with the strict-double-bound numbers of Cartesian products of graphs. We show that ζ (P n × P m) ≤ 2 n + 2 m − 4 for n , m ≥ 2 , ζ (C 4 n × C 4 m) ≤ 8 n m + 4 for n , m ≥ 1 , and ζ (C n × C m) ≤ 4 n + 4 m − 4 for n , m ≥ 3. [ABSTRACT FROM AUTHOR]