Incomparability Graphs of Lattices II
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 K n with two horns is realizable as Γ′(L). We also show that the complete graph K 3 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.
KeywordsIncomparability graph bipartite graph horn double star graph zero-divisor graph
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