Document

# Conflict-Free Coloring on Claw-Free Graphs and Interval Graphs

## File

LIPIcs.MFCS.2022.19.pdf
• Filesize: 0.68 MB
• 14 pages

## Acknowledgements

We would like to thank anonymous referees for helpful comments and corrections, and for suggesting an improvement in the proof of Lemma 6. We would also like to thank Saket Saurabh for helpful discussions.

## Cite As

Sriram Bhyravarapu, Subrahmanyam Kalyanasundaram, and Rogers Mathew. Conflict-Free Coloring on Claw-Free Graphs and Interval Graphs. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.MFCS.2022.19

## Abstract

A Conflict-Free Open Neighborhood coloring, abbreviated CFON^* coloring, of a graph G = (V,E) using k colors is an assignment of colors from a set of k colors to a subset of vertices of V(G) such that every vertex sees some color exactly once in its open neighborhood. The minimum k for which G has a CFON^* coloring using k colors is called the CFON^* chromatic number of G, denoted by χ_{ON}^*(G). The analogous notion for closed neighborhood is called CFCN^* coloring and the analogous parameter is denoted by χ_{CN}^*(G). The problem of deciding whether a given graph admits a CFON^* (or CFCN^*) coloring that uses k colors is NP-complete. Below, we describe briefly the main results of this paper. - For k ≥ 3, we show that if G is a K_{1,k}-free graph then χ_{ON}^*(G) = O(k²log Δ), where Δ denotes the maximum degree of G. Dębski and Przybyło in [J. Graph Theory, 2021] had shown that if G is a line graph, then χ_{CN}^*(G) = O(log Δ). As an open question, they had asked if their result could be extended to claw-free (K_{1,3}-free) graphs, which are a superclass of line graphs. Since it is known that the CFCN^* chromatic number of a graph is at most twice its CFON^* chromatic number, our result positively answers the open question posed by Dębski and Przybyło. - We show that if the minimum degree of any vertex in G is Ω(Δ/{log^ε Δ}) for some ε ≥ 0, then χ_{ON}^*(G) = O(log^{1+ε}Δ). This is a generalization of the result given by Dębski and Przybyło in the same paper where they showed that if the minimum degree of any vertex in G is Ω(Δ), then χ_{ON}^*(G)= O(logΔ). - We give a polynomial time algorithm to compute χ_{ON}^*(G) for interval graphs G. This answers in positive the open question posed by Reddy [Theoretical Comp. Science, 2018] to determine whether the CFON^* chromatic number can be computed in polynomial time on interval graphs. - We explore biconvex graphs, a subclass of bipartite graphs and give a polynomial time algorithm to compute their CFON^* chromatic number. This is interesting as Abel et al. [SIDMA, 2018] had shown that it is NP-complete to decide whether a planar bipartite graph G has χ_{ON}^*(G) = k where k ∈ {1, 2, 3}.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Combinatorial algorithms
• Mathematics of computing → Combinatorics
##### Keywords
• Conflict-free coloring
• Interval graphs
• Bipartite graphs
• Claw-free graphs

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Zachary. Abel, Victor. Alvarez, Erik D. Demaine, Sándor P. Fekete, Aman. Gour, Adam. Hesterberg, Phillip. Keldenich, and Christian. Scheffer. Conflict-free coloring of graphs. SIAM Journal on Discrete Mathematics, 32(4):2675-2702, 2018. URL: https://doi.org/10.1137/17M1146579.
2. Akanksha Agrawal, Pradeesha Ashok, Meghana M. Reddy, Saket Saurabh, and Dolly Yadav. FPT algorithms for conflict-free coloring of graphs and chromatic terrain guarding. CoRR, abs/1905.01822, 2019. URL: http://arxiv.org/abs/1905.01822.
3. Amotz Bar-Noy, Panagiotis Cheilaris, Svetlana Olonetsky, and Shakhar Smorodinsky. Online conflict-free colorings for hypergraphs. In Lars Arge, Christian Cachin, Tomasz Jurdziński, and Andrzej Tarlecki, editors, Automata, Languages and Programming, pages 219-230, Berlin, Heidelberg, 2007. Springer Berlin Heidelberg.
4. Sriram Bhyravarapu, Tim A. Hartmann, Subrahmanyam Kalyanasundaram, and I. Vinod Reddy. Conflict-free coloring: Graphs of bounded clique width and intersection graphs. In Combinatorial Algorithms - 32nd International Workshop, IWOCA 2021, Ottawa, ON, Canada, July 5-7, 2021, Proceedings, pages 92-106, 2021. URL: https://doi.org/10.1007/978-3-030-79987-8_7.
5. Sriram Bhyravarapu, Subrahmanyam Kalyanasundaram, and Rogers Mathew. A short note on conflict-free coloring on closed neighborhoods of bounded degree graphs. J. Graph Theory, 97(4):553-556, 2021. URL: https://doi.org/10.1002/jgt.22670.
6. Hans L. Bodlaender, Sudeshna Kolay, and Astrid Pieterse. Parameterized complexity of conflict-free graph coloring. CoRR, abs/1905.00305, 2019. URL: http://arxiv.org/abs/1905.00305.
7. Andreas Brandstädt and Vadim V. Lozin. On the linear structure and clique-width of bipartite permutation graphs. Ars Comb., 67, 2003.
8. Panagiotis Cheilaris. Conflict-free Coloring. PhD thesis, City University of New York, New York, NY, USA, 2009.
9. Ke Chen, Amos Fiat, Haim Kaplan, Meital Levy, Jiří Matoušek, Elchanan Mossel, János Pach, Micha Sharir, Shakhar Smorodinsky, Uli Wagner, and Emo Welzl. Online conflict-free coloring for intervals. SIAM J. Comput., 36(5):1342-1359, December 2006.
10. Michał Dębski and Jakub Przybyło. Conflict-free chromatic number versus conflict-free chromatic index. Journal of Graph Theory, 2021. URL: https://doi.org/10.1002/jgt.22743.
11. Reinhard Diestel. Graph theory 5th ed. Graduate texts in mathematics, 173, 2017.
12. Jessica A. Enright, Lorna Stewart, and Gábor Tardos. On list coloring and list homomorphism of permutation and interval graphs. SIAM J. Discret. Math., 28(4):1675-1685, 2014. URL: https://doi.org/10.1137/13090465X.
13. P. Erdős and L. Lovász. Problems and results on 3-chromatic hypergraphs and some related questions. Infinite and finite sets, 10:609-627, 1975.
14. Guy Even, Zvi Lotker, Dana Ron, and Shakhar Smorodinsky. Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellular networks. SIAM Journal on Computing, 33(1):94-136, January 2004.
15. Sándor P Fekete, Stephan Friedrichs, Michael Hemmer, Joseph BM Mitchell, and Christiane Schmidt. On the chromatic art gallery problem. In CCCG, 2014.
16. Sándor P. Fekete and Phillip Keldenich. Conflict-free coloring of intersection graphs. International Journal of Computational Geometry & Applications, 28(03):289-307, 2018.
17. Luisa Gargano and Adele Rescigno. Collision-free path coloring with application to minimum-delay gathering in sensor networks. Discrete Applied Mathematics, 157:1858-1872, April 2009. URL: https://doi.org/10.1016/j.dam.2009.01.015.
18. Luisa Gargano and Adele A. Rescigno. Complexity of conflict-free colorings of graphs. Theor. Comput. Sci., 566(C):39-49, February 2015. URL: https://doi.org/10.1016/j.tcs.2014.11.029.
19. Roman Glebov, Tibor Szabó, and Gábor Tardos. Conflict-free colouring of graphs. Combinatorics, Probability and Computing, 23(3):434-448, 2014.
20. Carolina Lucía Gonzalez and Felix Mann. On d-stable locally checkable problems on bounded mim-width graphs. CoRR, abs/2203.15724, 2022. URL: https://doi.org/10.48550/arXiv.2203.15724.
21. Frank Hoffmann, Klaus Kriegel, Subhash Suri, Kevin Verbeek, and Max Willert. Tight bounds for conflict-free chromatic guarding of orthogonal art galleries. Computational Geometry, 73:24-34, 2018.
22. Chaya Keller and Shakhar Smorodinsky. Conflict-free coloring of intersection graphs of geometric objects. In SODA, 2017.
23. Prasad Krishnan, Rogers Mathew, and Subrahmanyam Kalyanasundaram. Pliable index coding via conflict-free colorings of hypergraphs. In IEEE International Symposium on Information Theory, ISIT 2021, Melbourne, Australia, July 12-20, 2021, pages 214-219. IEEE, 2021. URL: https://doi.org/10.1109/ISIT45174.2021.9518120.
24. M. Mitzenmacher and E. Upfal. Probability and computing: Randomized algorithms and probabilistic analysis. Cambridge Univ Pr, 2005.
25. Vinodh P Vijayan and E. Gopinathan. Design of collision-free nearest neighbor assertion and load balancing in sensor network system. Procedia Computer Science, 70:508-514, December 2015. URL: https://doi.org/10.1016/j.procs.2015.10.092.
26. Janos Pach and Gábor Tardos. Conflict-free colourings of graphs and hypergraphs. Combinatorics, Probability and Computing, 18(5):819-834, 2009.
27. I. Vinod Reddy. Parameterized algorithms for conflict-free colorings of graphs. Theor. Comput. Sci., 745:53-62, 2018. URL: https://doi.org/10.1016/j.tcs.2018.05.025.
28. Shakhar Smorodinsky. Conflict-Free Coloring and its Applications, pages 331-389. Springer Berlin Heidelberg, Berlin, Heidelberg, 2013.