Abstract
The main goal of this paper is to formalize and explore a connection between chromatic properties of graphs with geometric representations and competitive analysis of on-line algorithms, which became apparent after the recent construction of triangle-free geometric intersection graphs with arbitrarily large chromatic number due to Pawlik et al. We show that on-line graph coloring problems give rise to classes of game graphs with a natural geometric interpretation. We use this concept to estimate the chromatic number of graphs with geometric representations by finding, for appropriate simpler graphs, on-line coloring algorithms using few colors or proving that no such algorithms exist.
We derive upper and lower bounds on the maximum chromatic number that rectangle overlap graphs, subtree overlap graphs, and interval filament graphs (all of which generalize interval overlap graphs) can have when their clique number is bounded. The bounds are absolute for interval filament graphs and asymptotic of the form (log logn)f(ω) for rectangle and subtree overlap graphs, where f(ω) is a polynomial function of the clique number and n is the number of vertices. In particular, we provide the first construction of geometric intersection graphs with bounded clique number and with chromatic number asymptotically greater than log logn.
We also introduce a concept of Kk-free colorings and show that for some geometric representations, K3-free chromatic number can be bounded in terms of clique number although the ordinary (K2-free) chromatic number cannot. Such a result for segment intersection graphs would imply a well-known conjecture that k-quasi-planar geometric graphs have linearly many edges.
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References
E. Ackerman: On the maximum number of edges in topological graphs with no four pairwise crossing edges, Discrete Comput. Geom. 41 (2009), 365–375.
P. K. Agarwal, B. Aronov, J. Pach, R. Pollack and M. Sharir: Quasiplanar graphs have a linear number of edges, Combinatorica 17 (1997), 1–9.
A. A. Ageev: A triangle-free circle graph with chromatic number 5, Discrete Math. 152 (1996), 295–298.
E. Asplund and B. Grünbaum: On a colouring problem, Math. Scand. 8 (1960), 181–188.
D. R. Bean: Effective coloration, J. Symb. Logic 41 (1976), 289–560.
J. P. Burling: On coloring problems of families of prototypes, PhD thesis, University of Colorado, Boulder, 1965.
J. Černý: Coloring circle graphs, Electron. Notes Discrete Math. 29 (2007), 457–461.
J. Chalopin, L. Esperet, Zh. Li and P. Ossona de Mendez: Restricted frame graphs and a conjecture of Scott, Electron. J. Combin. 23 (2016), P1.30.
J. Enright and L. Stewart: Subtree filament graphs are subtree overlap graphs, Inform. Process. Lett. 104 (2007), 228–232.
T. Erlebach and J. Fiala: On-line coloring of geometric intersection graphs, Comput. Geom. 23 (2002), 243–255.
S. Felsner: On-line chain partitions of orders, Theor. Comput. Sci. 175 (1997), 283–292.
J. Fox and J. Pach: Coloring Kk-free intersection graphs of geometric objects in the plane, European J. Combin. 33 (2012), 853–866.
J. Fox and J. Pach: Applications of a new separator theorem for string graphs, Combin. Prob. Comput. 23 (2014), 66–74.
F. Gavril: The intersection graphs of subtrees in trees are exactly the chordal graphs, J. Combin. Theory Ser. B 16 (1974), 47–56.
F. Gavril: Maximum weight independent sets and cliques in intersection graphs of filaments, Inform. Process. Lett. 73 (2000), 181–188.
M. C. Golumbic, D. Rotem and J. Urrutia: Comparability graphs and intersection graphs, Discrete Math. 43 (1983), 37–46.
A. Gyárfás: On the chromatic number of multiple interval graphs and overlap graphs, Discrete Math. 55 (1985), 161–166. Corrigendum: Discrete Math. 62 (1986), 333.
A. Gyárfás and J. Lehel: On-line and first fit colorings of graphs, J. Graph Theory 12 (1988), 217–227.
C. Hendler: Schranken für Färbungs- und Cliquenüberdeckungszahl geometrisch repräsentierbarer Graphen, Master’s thesis, Freie Universität Berlin, 1998.
H. A. Kierstead and W. T. Trotter: An extremal problem in recursive combinatorics, in: F. Hoffman (ed.), 3rd Southeastern International Conference on Combinatorics, Graph Theory, and Computing (CGTC 1981), vol. 33 of Congressus Numerantium, 143–153, Utilitas Math. Pub., Winnipeg, 1981.
A. Kostochka: On upper bounds for the chromatic numbers of graphs, Trudy Inst. Mat. 10 (1988), 204–226 (in Russian).
A. Kostochka: Coloring intersection graphs of geometric figures with a given clique number, in: J. Pach (ed.), Towards a Theory of Geometric Graphs, vol. 342 of Contemp. Math., 127–138, AMS, Providence, 2004.
A. Kostochka and J. Kratochvíl: Covering and coloring polygon-circle graphs, Discrete Math. 163 (1997), 299–305.
A. Kostochka and K. Milans: Coloring clean and K4-free circle graphs, in: J. Pach (ed.), Thirty Essays on Geometric Graph Theory, 399–414, Springer, New York, 2012.
T. Krawczyk, A. Pawlik and B. Walczak: Coloring triangle-free rectangle overlap graphs with O(log logn) colors, Discrete Comput. Geom. 53 (2015), 199–220.
L. Lovász: Perfect graphs, in: L. W. Beineke and R. J. Wilson (eds.), Selected Topics in Graph Theory, vol. 2, 55–87, Academic Press, London, 1983.
J. Pach, R. Radoičić and G. Tóth: Relaxing planarity for topological graphs, in: E. Győri, Gy. O. H. Katona and L. Lovász (eds.), More Graphs, Sets and Numbers, vol. 15 of Bolyai Soc. Math. Stud., 285–300, Springer, Berlin, 2006.
J. Pach, F. Shahrokhi and M. Szegedy: Applications of the crossing number, Algorithmica 16 (1996), 111–117.
A. Pawlik, J. Kozik, T. Krawczyk, M. Lasoń, P. Micek, W. T. Trotter and B. Walczak: Triangle-free geometric intersection graphs with large chromatic number, Discrete Comput. Geom. 50 (2013), 714–726.
A. Pawlik, J. Kozik, T. Krawczyk, M. Lasoń, P. Micek, W. T. Trotter and B. Walczak: Triangle-free intersection graphs of line segments with large chromatic number, J. Combin. Theory Ser. B 105 (2014), 6–10.
M. Pergel: Recognition of polygon-circle graphs and graphs of interval filaments is NP-complete, in: A. Brandstädt, D. Kratsch and H. Müller (eds.), 33rd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2007), vol. 4769 of Lecture Notes Comput. Sci., 238–247, Springer, Berlin, 2007.
A. Rok and B. Walczak: Outerstring graphs are χ-bounded, in: S.-W. Cheng and O. Devillers (eds.), 30th Annual Symposium on Computational Geometry (SoCG 2014), 136–143, ACM, New York, 2014.
D. D. Sleator and R. E. Tarjan: A data structure for dynamic trees, J. Comput. System Sci. 26 (1983), 362–391.
A. Suk and B. Walczak: New bounds on the maximum number of edges in k-quasiplanar graphs, Comput. Geom. 50 (2015), 24–33.
P. Valtr: On geometric graphs with no k pairwise parallel edges, Discrete Comput. Geom. 19 (1998), 461–469.
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A preliminary version of this paper appeared as: Coloring relatives of interval overlap graphs via on-line games, in: J. Esparza, P. Fraigniaud, T. Husfeldt and E. Koutsoupias (eds.), 41st International Colloquium on Automata, Languages, and Programming (ICALP 2014), part I, vol. 8572 of Lecture Notes Comput. Sci., Springer, Berlin, 2014.
Tomasz Krawczyk and Bartosz Walczak were partially supported by National Science Center of Poland grant 2011/03/B/ST6/01367. Bartosz Walczak was partially supported by Swiss National Science Foundation grant 200020-144531.
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Krawczyk, T., Walczak, B. On-Line Approach to Off-Line Coloring Problems on Graphs with Geometric Representations. Combinatorica 37, 1139–1179 (2017). https://doi.org/10.1007/s00493-016-3414-x
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DOI: https://doi.org/10.1007/s00493-016-3414-x