Abstract
The coloring of disk graphs is motivated by the frequency assignment problem. In 1998, Malesińska et al. introduced double disk graphs as their generalization. They showed that the chromatic number of a double disk graph \(G\) is at most \(33\,\omega (G) - 35\), where \(\omega (G)\) denotes the size of a maximum clique in \(G\). Du et al. improved the upper bound to \(31\,\omega (G) - 1\). In this paper we decrease the bound substantially; namely we show that the chromatic number of \(G\) is at most \(15\,\omega (G) - 14\).
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References
Aardal, K.I., van Hoesel, S.P.M., Koster, A.M.C.A., Mannino, C., Sassano, A.: Models and solution techniques for frequency assignment problems. Ann. Oper. Res. 153, 79–129 (2007)
Balasundaram, B., Butenko, S.: Optimization problems in unit-disk graphs. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, pp. 2832–2844. Springer, Dordrecht (2009)
Caragiannis, I., Fishkin, A.V., Kaklamanis, C., Papaioannou, E.: A tight bound for online colouring of disk graphs. Theor. Comput. Sci. 384(2–3), 152–160 (2007)
Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discret. Math. 86(1–3), 165–177 (1990)
Du, H., Jia, X., Li, D., Wu, W.: Coloring of double disk graphs. J. Global Optim. 28(1), 115–119 (2004)
Fiala, J., Fishkin, A.V., Fomin, F.: On distance constrained labeling of disk graphs. Theor. Comput. Sci. 326(1–3), 261–292 (2004)
Fishkin, A.: Disk graphs: a short survey. In: Solis-Oba, R., Jansen, K. (eds.) Approximation and Online Algorithms. Lecture Notes in Computer Science, vol. 2909, pp. 260–264. Springer, Berlin Heidelberg (2004)
Gräf, A., Stumpf, M., Weißenfels, G.: On coloring unit disk graphs. Algorithmica 20(3), 277–293 (1998)
Havet, F., Kang, R.J., Sereni, J.S.: Improper coloring of unit disk graphs. Networks 54(3), 150–164 (2009)
Kang, R.J., Müller, T., Sereni, J.S.: Improper colouring of (random) unit disk graphs. Discret. Math. 308(8), 1438–1454 (2008)
Kanj, I.A., Wiese, A., Zhang, F.: Local algorithms for edge colorings in UDGs. Theor. Comput. Sci. 412(35), 4704–4714 (2011)
Lev-Tov, N., Peleg, D.: Conflict-free coloring of unit disks. Discret. Appl. Math. 157(7), 1521–1532 (2009)
Lick, D.R., White, A.T.: k-Degenerate graphs. Can. J. Math. 22, 1082–1096 (1970)
Malesińska, E., Piskorz, S., Weißenfels, G.: On the chromatic number of disk graphs. Networks 32(1), 13–22 (1998)
Matula, D.W., Beck, L.L.: Smallest-last ordering and clustering and graph coloring algorithms. J. ACM 30, 417–427 (1983)
Marathe, M.V., Breu, H., Hunt III, H.B., Ravi, S.S., Rosenkrantz, D.J.: Simple heuristics for unit disk graphs. Networks 25(2), 59–68 (1995)
Peeters, R.: On Coloring \(j\)-unit Sphere Graphs. FEW 512, Department of Economics, Tilburg University, Tilburg (1991)
Acknowledgments
This work is partially supported by ARRS Program P1-0383 and by Creative Core FISNM-3330-13-500033. Additionally, the third author was partially financed by the project NEXLIZ - CZ.1.07/2.3.00/30.0038, which is co-financed by the European Social Fund and the state budget of the Czech Republic, and the fourth author was supported in part by Slovak VEGA Grant No. 1/0652/12, by the Slovak Science and Technology Assistance Agency under the contract APVV-0023-10, and the Project implementation: University Science Park TECHNICOM for Innovation Applications Supported by Knowledge Technology, ITMS: 26220220182, supported by the Research & Development Operational Programme funded by the ERDF.
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Kranjc, J., Lužar, B., Mockovčiaková, M. et al. Note on coloring of double disk graphs. J Glob Optim 60, 793–799 (2014). https://doi.org/10.1007/s10898-014-0221-z
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DOI: https://doi.org/10.1007/s10898-014-0221-z