Abstract
In an undirected graph G, a conflict-free coloring with respect to open neighborhoods (denoted by CFON coloring) is an assignment of colors to the vertices such that every vertex has a uniquely colored vertex in its open neighborhood. The minimum number of colors required for a CFON coloring of G is the CFON chromatic number of G, denoted by \(\chi _{ON}(G)\).
The decision problem that asks whether \(\chi _{ON}(G)\le k\) is NP-complete. Structural as well as algorithmic aspects of this problem have been well studied. We obtain the following results for \(\chi _{ON}(G)\):
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Bodlaender, Kolay and Pieterse [WADS 2019] showed the upper bound \(\chi _{ON}(G)\le \mathsf{fvs}(G) + 3\), where \(\mathsf{fvs}(G)\) denotes the size of a minimum feedback vertex set of G. We show the improved bound of \(\chi _{ON}(G)\le \mathsf{fvs}(G) + 2\), which is tight, thereby answering an open question in the above paper.
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We study the relation between \(\chi _{ON}(G)\) and the pathwidth of the graph G, denoted \(\mathsf{pw}(G)\). The above paper from WADS 2019 showed the upper bound \(\chi _{ON}(G)\le 2\mathsf{tw}(G) + 1\) where \(\mathsf{tw}(G)\) stands for the treewidth of G. This implies an upper bound of \(\chi _{ON}(G)\le 2\mathsf{pw}(G) + 1\). We show an improved bound of
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We prove new bounds for \(\chi _{ON}(G)\) with respect to the structural parameters neighborhood diversity and distance to cluster, improving the existing results of Gargano and Rescigno [Theor. Comput. Sci. 2015] and Reddy [Theor. Comput. Sci. 2018], respectively. Furthermore, our techniques also yield improved bounds for the closed neighborhood variant of the problem.
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We also study the partial coloring variant of the CFON coloring problem, which allows vertices to be left uncolored. Let \(\chi ^*_{ON}(G)\) denote the minimum number of colors required to color G as per this variant. Abel et al. [SIDMA 2018] showed that \(\chi ^*_{ON}(G)\le 8\) when G is planar. They asked if fewer colors would suffice for planar graphs. We answer this question by showing that \(\chi ^*_{ON}(G)\le 5\) for all planar G. This approach also yields the bound \(\chi ^*_{ON}(G)\le 4\) for all outerplanar G.
All our bounds are a result of constructive algorithmic procedures.
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Notes
- 1.
In the case where X is a special bag that introduces two vertices, at most one of the two introduced vertices can be part of an expensive subset.
- 2.
The vertex \(v'_i\) may or may not be the same as \(v_i\).
References
Even, G., Lotker, Z., Ron, D., Smorodinsky, S.: Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellular networks. SIAM J. Comput. 33(1), 94–136 (2004). https://doi.org/10.1137/S0097539702431840
Smorodinsky, S.: Conflict-free coloring and its applications. In: Bárány, I., Böröczky, K.J., Tóth, G.F., Pach, J. (eds.) Geometry—Intuitive, Discrete, and Convex. BSMS, vol. 24, pp. 331–389. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-41498-5_12
Abel, Z., et al.: Conflict-free coloring of graphs. SIAM J. Discret. Math. 32(4), 2675–2702 (2018). https://doi.org/10.1137/17M1146579
Gargano, L., Rescigno, A.A.: Complexity of conflict-free colorings of graphs. Theor. Comput. Sci. 566(C), 39–49 (2015). https://doi.org/10.1016/j.tcs.2014.11.029
Reddy, I.V.: Parameterized algorithms for conflict-free colorings of graphs. Theor. Comput. Sci. 745, 53–62 (2018). https://doi.org/10.1016/j.tcs.2018.05.025
Bodlaender, H.L., Kolay, S., Pieterse, A.: Parameterized complexity of conflict-free graph coloring. In: Friggstad, Z., Sack, J.-R., Salavatipour, M.R. (eds.) WADS 2019. LNCS, vol. 11646, pp. 168–180. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-24766-9_13
Agrawal, A., Ashok, P., Reddy, M.M., Saurabh, S., Yadav, D.: FPT algorithms for conflict-free coloring of graphs and chromatic terrain guarding. CoRR abs/1905.01822. arXiv:1905.01822 (2019)
Keller, C., Smorodinsky, S.: Conflict-free coloring of intersection graphs of geometric objects. Discret. Comput. Geom., 1–26 (2019). https://doi.org/10.1007/s00454-019-00097-8
Fekete, S.P., Keldenich, P.: Conflict-free coloring of intersection graphs. Int. J. Comput. Geom. Appl. 28(03), 289–307 (2018). https://doi.org/10.1142/S0218195918500085
Chen, K.: Online conflict-free coloring for intervals. SIAM J. Comput. 36(5), 1342–1359 (2006). https://doi.org/10.1137/S0097539704446682
Pach, J., Tardos, G.: Conflict-free colourings of graphs and hypergraphs. Comb. Probab. Comput. 18(5), 819–834 (2009). https://doi.org/10.1017/S0963548309990290
Cheilaris, P.: Conflict-free coloring, Ph.D. thesis, New York, NY, USA (2009)
Bhyravarapu, S., Kalyanasundaram, S.: Combinatorial bounds for conflict-free coloring on open neighborhoods. CoRR abs/2007.05585. arXiv:2007.05585 (2020)
Cygan, M., et al.: Parameterized Algorithms, vol. 1. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21275-3
Robertson, N., Sanders, D., Seymour, P., Thomas, R.: The four-colour theorem. J. Comb. Theor. Ser. B 70(1), 2–44 (1997). https://doi.org/10.1006/jctb.1997.1750
Acknowledgments
We would like to thank I. Vinod Reddy for suggesting the problem, Rogers Mathew and N. R. Aravind for helpful discussions and the anonymous reviewer who pointed out an issue with the proof of Theorem 6.
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Bhyravarapu, S., Kalyanasundaram, S. (2020). Combinatorial Bounds for Conflict-Free Coloring on Open Neighborhoods. In: Adler, I., Müller, H. (eds) Graph-Theoretic Concepts in Computer Science. WG 2020. Lecture Notes in Computer Science(), vol 12301. Springer, Cham. https://doi.org/10.1007/978-3-030-60440-0_1
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