Lattices, whose incomparability graphs have horns

In this paper, we study some graphs which are realizable and some which are not realizable as the incomparability graph (denoted by Γ′(L)Γ′(L)) of a lattice L   with at least two atoms. We prove that the complete graph KnKn with two horns is realizable as Γ′(L)Γ′(L). We show that the complete graph K3K3 with three horns is not realizable as Γ′(L)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 one horn and with two horns to be realizable as Γ′(L)Γ′(L) for some lattice L.