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 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.
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Afkhami, M., Khashyarmanesh, K.: The cozero divisor graph of a commutative ring. Southeast Asian Bull. Math. 35, 753–762 (2011)
Bollobas, B., Rival, I.: The maximal size of the covering graph of a lattice. Algebra Universalis 9, 371–373 (1979)
Bresar, B., Changat, M., Klavzar, S., Kovse, M., Mathews, J., Mathews, A.: Cover - incomparability graphs of posets. Order 25, 335–347 (2008)
Duffus, D., Rival, I.: Path lengths in the covering graph. Discrete Math. 19, 139–158 (1977)
Filipov, N.D.: Comparability graphs of partially ordered sets of different types. Collq. Maths. Soc. Janos Bolyai 33, 373–380 (1980)
Gedenova, E.: Lattices, whose covering graphs are s- graphs. Colloq. Math. Soc. Janos Bolyai 33, 407–435 (1980)
Grätzer, G.: General Lattice Theory. Birkhauser, Basel (1998)
Harary, F.: Graph Theory, Narosa, New Delhi (1988)
Nimbhorkar, S.K., Wasadikar, M.P., Demeyer, L.: Coloring of meet semilattices. Ars Combin. 84, 97–104 (2007)
Nimbhorkar, S.K., Wasadikar, M.P., Pawar, M.M.: Coloring of lattices. Math. Slovaca 60, 419–434 (2010)
Wasadikar, M., Survase, P.: Some properties of graphs derived from lattices. Bull. Calcutta Math. Soc. 104, 125–138 (2012)
Wasadikar, M., Survase, P.: Incomparability Graphs of Lattices. In: Balasubramaniam, P., Uthayakumar, R. (eds.) ICMMSC 2012. CCIS, vol. 283, pp. 78–85. Springer, Heidelberg (2012)
Wasadikar, M., Survase, P.: The zero-divisor graph of a meet-semilattice. J. Combinatorial Math. and Combinatorial Computing (accepted)
West, D.B.: Introduction to Graph Theory. Prentice-Hall, New Delhi (1996)
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Wasadikar, M., Survase, P. (2012). Incomparability Graphs of Lattices II. In: Arumugam, S., Smyth, W.F. (eds) Combinatorial Algorithms. IWOCA 2012. Lecture Notes in Computer Science, vol 7643. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35926-2_18
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DOI: https://doi.org/10.1007/978-3-642-35926-2_18
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