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Abstract. Let L be a finite lattice with atleast two atoms and W (L) = {x | there exists y ∈ L such that x y }. The incomparability graph of L, denoted by Γ (L), is a graph with vertex set W (L) and two distinct vertices a, b ∈ W (L) are adjacent if and only if they are incomparable. In this paper, we study the incomparability graphs of lattices. We prove that, a disconnected graph is a graph of a lattice L if and only if L is of the form L1 L2. We prove that, Γ (L) cannot be an n-gon for any n ≥ 5. Some properties of incomparability graphs are obtained.

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.