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11 - Dimension and Cut Vertices: An Application of Ramsey Theory

Published online by Cambridge University Press:  25 May 2018

William T. Trotter
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA
Bartosz Walczak
Affiliation:
Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków 30-348, Poland
Ruidong Wang
Affiliation:
Blizzard Entertainment, Irvine, CA 92618, USA
Steve Butler
Affiliation:
Iowa State University
Joshua Cooper
Affiliation:
University of South Carolina
Glenn Hurlbert
Affiliation:
Virginia Commonwealth University
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Summary

Abstract

Motivated by quite recent research involving the relationship between the dimension of a poset and graph-theoretic properties of its cover graph, we show that for every, if P is a poset and the dimension of a subposet B of P is at most d whenever the cover graph of B is a block of the cover graph of P, then the dimension of P is at most d + 2.We also construct examples that show that this inequality is best possible. We consider the proof of the upper bound to be fairly elegant and relatively compact. However, we know of no simple proof for the lower bound, and our argument requires a powerful tool known as the Product Ramsey Theorem. As a consequence, our constructions involve posets of enormous size.

Introduction

We assume that the reader is familiar with basic notation and terminology for partially ordered sets (here we use the short term posets), including chains and antichains, minimal and maximal elements, linear extensions, order diagrams, and cover graphs. Extensive background information on the combinatorics of posets can be found in [17, 18].

We will also assume that the reader is familiar with basic concepts of graph theory, including the following terms: connected and disconnected graphs, components, cut vertices, and k-connected graphs for an integer. Recall that when G is a connected graph, a connected induced subgraph H of G is called a block of G when H is 2-connected and there is no subgraph of G which contains H as a proper subgraph and is also 2-connected.

Here are the analogous concepts for posets. A poset P is said to be connected if its cover graph is connected. A subposet B of P is said to be convex if yB whenever x, zB and x < y < z in P. Note that when B is a convex subposet of P, the cover graph of B is an induced subgraph of the cover graph of P. A convex subposet B of P is called a component of P when the cover graph of B is a component of the cover graph of P. A convex subposet B of P is called a block of P when the cover graph of B is a block in the cover graph of P.

Type
Chapter
Information
Connections in Discrete Mathematics
A Celebration of the Work of Ron Graham
, pp. 187 - 199
Publisher: Cambridge University Press
Print publication year: 2018

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References

1. Csaba, Biro, Mitchel T., Keller, and Stephen J., Young. Posets with cover graph of pathwidth two have bounded dimension. Order 33, no. 2 (2016), 195–212.Google Scholar
2. Csaba, Biro, Peter, Hamburger, Attila, Por, and William T., Trotter. Forcing posets with large dimension to contain large stand ard examples. Graphs Combin. 32, no. 3 (2016), 861–880.Google Scholar
3. Ben, Dushnik and Edwin W., Miller. Partially ordered sets. Amer. J. Math. 63, no. 3 (1941), 600–610.Google Scholar
4. Stefan, Felsner, Peter C., Fishburn, and William T., Trotter. Finite three dimensional partial orders which are not sphere orders. Discrete Math. 201, no. 1–3 (1999), 101–132.Google Scholar
5. Stefan, Felsner, William T., Trotter, and Veit, Wiechert. The dimension of posets with planar cover graphs. Graphs Combin. 31, no. 4 (2015), 927–939.Google Scholar
6. Peter C., Fishburn and Ronald L., Graham. Lexicographic Ramsey theory. J. Combin. Theory Ser. A 62, no. 2 (1993), 280–298.Google Scholar
7. Ronald L., Graham, Bruce L., Rothschild, and Joel H., Spencer. Ramsey Theory, 2nd edn. John Wiley & Sons, New York, 1990.Google Scholar
8. Gwenael, Joret, Piotr, Micek, Kevin G., Milans, William T., Trotter, Bartosz, Walczak, and Ruidong, Wang. Tree-width and dimension. Combinatorica 36, no. 4 (2016), 431–450.Google Scholar
9. Gwenael, Joret, Piotr, Micek, William T., Trotter, Ruidong, Wang, and Veit, Wiechert. On the dimension of posets with cover graphs of treewidth 2. Order 34, no. 2 (2017), 185–234.Google Scholar
10. Gwenael, Joret, Piotr, Micek, and Veit, Wiechert. Sparsity and dimension. Combinatorica, in press, doi: 10.1007/s00493-017-3638-4.CrossRef
11. David, Kelly. The 3-irreducible partially ordered sets. Canad. J. Math. 29 (1977), 367–383.Google Scholar
12. Piotr, Micek and Veit, Wiechert. Topological minors of cover graphs and dimension. J. Graph Theory 86, no. 3 (2017), 295–314.Google Scholar
13. Noah, Streib and William T., Trotter. Dimension and height for posets with planar cover graphs. Eur. J. Combin. 35 (2014), 474–489.Google Scholar
14. William T., Trotter. Irreducible posets with large height exist. J. Combin. Theory Ser. A 17, no. 3 (1974), 337–344.Google Scholar
15. William T., Trotter. Inequalities in dimension theory for posets. Proc. Amer. Math. Soc. 47, no. 2 (1975), 311–316.Google Scholar
16. William T., Trotter. Combinatorial problems in dimension theory for partially ordered sets. In Problemes Combinatoires et Theorie des Graphes, Vol. 260 of Colloques Internationeaux C.N.R.S., pp. 403–406, Editions du C.N.R.S., Paris, 1978.Google Scholar
17. William T., Trotter. Combinatorics and Partially Ordered Sets: Dimension Theory. Johns Hopkins University Press, Baltimore, 1992.Google Scholar
18. William T., Trotter. Partially ordered sets. In Ronald L., Graham, Martin, Grotschel, and Laszlo, Lovasz (eds.), Hand book of Combinatorics, Vol. I, pp. 433–480, North- Holland, Amsterdam, 1995.Google Scholar
19. William T., Trotter and John I., Moore. Characterization problems for graphs, partially ordered sets, lattices, and families of sets. Discrete Math. 16, no. 4 (1976), 361–381.Google Scholar
20. William T., Trotter and John I., Moore. The dimension of planar posets. J. Combin. Theory Ser. B 22, no. 1 (1977), 54–67.Google Scholar
21. William T., Trotter and Ruidong, Wang. Dimension and matchings in comparability and incomparability graphs. Order 33, no. 1 (2016), 101–119. 22. Bartosz Walczak. Minors and dimension. J. Combin. Theory Ser. B 122 (2017), 668–689.Google Scholar
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