There is a natural way to associate with a poset P a hypergraph HP, called the hypergraph of incomparable pairs, so that the dimension of P is the chromatic number of HP. The ordinary graph GP of incomparable pairs determined by the edges in HP of size 2 can have chromatic number substantially less than HP. We give a new proof of the fact that the dimension of P is 2 if and only if GP is bipartite. We also show that for each t ≥ 2, there exists a poset P t for which the chromatic number of the graph of incomparable pairs of P t is at most 3 t − 4, but the dimension of P t is at least (3 / 2)t − 1. However, it is not known whether there is a function f: N→N so that if P is a poset and the graph of incomparable pairs has chromatic number at most t, then the dimension of P is at most f(t).
Unable to display preview. Download preview PDF.
- 1.Cogis, O. (1980) La dimension Ferrers des graphes orientés, Thèse, Université Pierre et Marie Curie, Paris.Google Scholar
- 2.Felsner, S. and Trotter, W. T. The Dimension of the Adjacency Poset of a Planar Graph, in preparation.Google Scholar
- 3.Füredi, Z., Hajnal, P., Rödl, V. and Trotter, W. T. (1991) Interval orders and shift graphs, in A. Hajnal and V. T. Sos (eds), Sets, Graphs and Numbers, Colloq. Math. Soc. Janos Bolyai 60, pp. 297–313.Google Scholar
- 4.Trotter, W. T. (1992) Combinatorics and Partially Ordered Sets: Dimension Theory, The Johns Hopkins University Press, Baltimore, MD.Google Scholar
- 5.Trotter, W. T. (1995) Partially ordered sets, in R. L. Graham, M. Grötschel, L. Lovász (eds), Handbook of Combinatorics, Vol. I, Elsevier, Amsterdam, pp. 433–480.Google Scholar
- 6.Trotter, W. T. (1996) Graphs and partially ordered sets, Congressus Numerantium 116, 253–278.Google Scholar
- 7.Trotter, W. T. (1997) New perspectives on interval orders and interval graphs, in R. A. Bailey (ed.), Surveys in Combinatorics, London Math. Soc. Lecture Note Ser. 241, pp. 237–286.Google Scholar
- 8.Trotter, W. T. and Moore J. I. (1977) The dimension of planar posets, J. Comb. Theory B 21, 51–67.Google Scholar
- 9.Yannakakis, M. (1982) On the complexity of the partial order dimension problem, SIAM J. Alg. Discr. Meth. 3, 351–358.Google Scholar