On the connectivity of the disjointness graph of segments of point sets in general position in the plane. (English) Zbl 1515.05099
Summary: Let \(P\) be a set of \(n\geq 3\) points in general position in the plane. The edge disjointness graph \(D(P)\) of \(P\) is the graph whose vertices are all the closed straight line segments with endpoints in \(P\), two of which are adjacent in \(D(P)\) if and only if they are disjoint. We show that the connectivity of \(D(P)\) is at least \(\binom{\lfloor\frac{n-2}{2}\rfloor}{2}+\binom{\lceil\frac{n-2}{2}\rceil}{2} \), and that this bound is tight for each \(n\geq 3\).
MSC:
| 05C40 | Connectivity |
| 52C35 | Arrangements of points, flats, hyperplanes (aspects of discrete geometry) |
References:
| [1] | M. O. Albertson and D. L. Boutin. Using determining sets to distinguish kneser graphs. Electron. J. Combin., 2007. 14(1): Research paper 20 (Electronic). · Zbl 1114.05089 |
| [2] | G. Araujo, A. Dumitrescu, F. Hurtado, M. Noy, and J. Urrutia. On the chromatic number of some geo-metric type kneser graphs. Comput. Geom., 32(1):59-69, 2005. · Zbl 1067.05023 |
| [3] | I. Bárány. A short proof of kneser’s conjecture. J. Combin. Theory Ser. A, 25:325-326, 1978. · Zbl 0404.05028 |
| [4] | Y.-C. Chen. Kneser graphs are hamiltonian for n ≥ 3k. J. Combin. Theory Ser. B, 80:69-79, 2000. · Zbl 1024.05055 |
| [5] | G. B. Ekinci and J. B. Gauci. The super-connectivity of kneser graphs. Discuss. Math. Graph Theory, 39: 5-11, 2019. · Zbl 1401.05164 |
| [6] | R. Fabila-Monroy and D. R. Wood. The chromatic number of the convex segment disjointness graph. Computational geometry, 7579 of Lecture Notes in Comput. Sci.:79-84, 2011. Springer, Cham,. · Zbl 1375.68131 |
| [7] | R. Fabila-Monroy, C. Hidalgo-Toscano, J. Leaños, and M. Lomelí-Haro. The chromatic number of the disjointness graph of the double chain. Discrete Mathematics and Theoretical Computer Science, 22:1, 2020. · Zbl 1450.05026 |
| [8] | J. Jonsson. The exact chromatic number of the convex segment disjointness graph. 2011. |
| [9] | M. Kneser. Jahresbericht der deutschen mathematiker-vereinigung. 58:27, 1956. Aufgabe 360. |
| [10] | L. Lovász. Kneser’s conjecture, chromatic number, and homotopy. J. Combin. Theory Ser. A, 25:319-324, 1978. · Zbl 0418.05028 |
| [11] | J. Matousěk. Using the Borsuk-Ulam theorem. Universitext. Springer-Verlag, Berlin, 2003. ISBN 3-540-00362-2. Lectures on topological methods in combinatorics and geometry, Written in cooperation with Anders Björner and Günter M. Ziegler. · Zbl 1016.05001 |
| [12] | J. Pach and I. Tomon. On the chromatic number of disjointness graphs of curves. 2019. In 35th Interna-tional Symposium on Computational Geometry (SoCG 2019). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. |
| [13] | J. Pach, G. Tardos, and G. Tóth. Disjointness graphs of segments. in: Boris Aronov and Matthew J.Katz (eds.), 77 of Leibniz International Proceedings in Informatics (LIPIcs), Leibniz-Zentrum für Informatik, Dagstuhl:59:1-15, 2017. 33th International Symposium on Computational Geometry (SoCG 2017). · Zbl 1432.68365 |
| [14] | P. Reinfeld. Chromatic polynomials and the spectrum of the kneser graph. 2000. tech. rep., London School of Economics, 2000. LSE-CDAM-2000-02. |
| [15] | M. Valencia-Pabon and J. C. Vera. On the diameter of kneser graphs. Discrete Math., 305:383-385, 2005. · Zbl 1100.05030 |
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