Incomparability Graphs of Lattices II

Abstract

Abstract. In this paper, we study some graphs which are realizable and some which are not realizable as the incomparability graph (denoted by Γ (L)) of a lattice L with at least two atoms. We prove that for n ≥ 4, the complete graph Kn with two horns is realizable as Γ (L). We also show that the complete graph K3 with three horns emanating from each of the three vertices is not realizable as Γ (L), however it is realizable as the zero-divisor graph of L. Also we give a necessary and sufficient condition for a complete bipartite graph with two horns to be realizable as Γ (L) for some lattice L.

Incomparability Graphs of Lattices

“…It is shown in [10] that there is no lattice L whose zero divisor graph is a double star graph. However a double star graph can be realizable as Γ (L) for a lattice L, see It is shown in [10] that diamΓ (L) ≤ 3. However for Γ (L) we have the following remark.…”