Document

# Linear Transformations Between Colorings in Chordal Graphs

## File

LIPIcs.ESA.2019.24.pdf
• Filesize: 0.64 MB
• 15 pages

## Cite As

Nicolas Bousquet and Valentin Bartier. Linear Transformations Between Colorings in Chordal Graphs. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ESA.2019.24

## Abstract

Let k and d be such that k >= d+2. Consider two k-colorings of a d-degenerate graph G. Can we transform one into the other by recoloring one vertex at each step while maintaining a proper coloring at any step? Cereceda et al. answered that question in the affirmative, and exhibited a recolouring sequence of exponential length. If k=d+2, we know that there exists graphs for which a quadratic number of recolorings is needed. And when k=2d+2, there always exists a linear transformation. In this paper, we prove that, as long as k >= d+4, there exists a transformation of length at most f(Delta) * n between any pair of k-colorings of chordal graphs (where Delta denotes the maximum degree of the graph). The proof is constructive and provides a linear time algorithm that, given two k-colorings c_1,c_2 computes a linear transformation between c_1 and c_2.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Graph theory
##### Keywords
• graph recoloring
• chordal graphs

## Metrics

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

## References

1. Hans-Joachim Böckenhauer, Elisabet Burjons, Martin Raszyk, and Peter Rossmanith. Reoptimization of Parameterized Problems. arXiv e-prints, 2018. URL: http://arxiv.org/abs/1809.10578.
2. M. Bonamy, M. Johnson, I. Lignos, V. Patel, and D. Paulusma. Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs. Journal of Combinatorial Optimization, pages 1-12, 2012. URL: https://doi.org/10.1007/s10878-012-9490-y.
3. Marthe Bonamy and Nicolas Bousquet. Token Sliding on Chordal Graphs. In Graph-Theoretic Concepts in Computer Science - 43rd International Workshop, WG 2017, pages 127-139, 2017.
4. Marthe Bonamy and Nicolas Bousquet. Recoloring graphs via tree decompositions. Eur. J. Comb., 69:200-213, 2018.
5. P. Bonsma and L. Cereceda. Finding Paths Between Graph Colourings: PSPACE-Completeness and Superpolynomial Distances. In MFCS, volume 4708 of Lecture Notes in Computer Science, pages 738-749, 2007.
6. Prosenjit Bose, Anna Lubiw, Vinayak Pathak, and Sander Verdonschot. Flipping edge-labelled triangulations. Comput. Geom., 68:309-326, 2018.
7. Nicolas Bousquet, Tatsuhiko Hatanaka, Takehiro Ito, and Moritz Mühlenthaler. Shortest Reconfiguration of Matchings. CoRR, abs/1812.05419, 2018. URL: http://arxiv.org/abs/1812.05419.
8. Nicolas Bousquet and Marc Heinrich. A polynomial version of Cereceda’s conjecture. arXiv e-prints, 2019. URL: http://arxiv.org/abs/1903.05619.
9. Nicolas Bousquet and Arnaud Mary. Reconfiguration of Graphs with Connectivity Constraints. In Approximation and Online Algorithms - 16th International Workshop, WAOA 2018, pages 295-309, 2018. URL: https://doi.org/10.1007/978-3-030-04693-4_18.
10. Nicolas Bousquet and Guillem Perarnau. Fast recoloring of sparse graphs. Eur. J. Comb., 52:1-11, 2016.
11. L. Cereceda. Mixing Graph Colourings. PhD thesis, London School of Economics and Political Science, 2007.
12. L. Cereceda, J. van den Heuvel, and M. Johnson. Mixing 3-colourings in bipartite graphs. Eur. J. Comb., 30(7):1593-1606, 2009. URL: https://doi.org/10.1016/j.ejc.2009.03.011.
13. L. Cereceda, J. van den Heuvel, and M. Johnson. Finding paths between 3-colorings. Journal of Graph Theory, 67(1):69-82, 2011. URL: https://doi.org/10.1002/jgt.20514.
14. Sitan Chen, Michelle Delcourt, Ankur Moitra, Guillem Perarnau, and Luke Postle. Improved Bounds for Randomly Sampling Colorings via Linear Programming. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, pages 2216-2234, 2019. URL: https://doi.org/10.1137/1.9781611975482.134.
15. R. Diestel. Graph Theory, volume 173 of Graduate Texts in Mathematics. Springer-Verlag, Heidelberg, third edition, 2005.
16. M. Dyer, A. D. Flaxman, A. M Frieze, and E. Vigoda. Randomly coloring sparse random graphs with fewer colors than the maximum degree. Random Structures & Algorithms, 29(4):450-465, 2006.
17. Carl Feghali. Paths between colourings of graphs with bounded tree-width. Information Processing Letters, 144, December 2018. URL: https://doi.org/10.1016/j.ipl.2018.12.006.
18. Carl Feghali, Matthew Johnson, and Daniël Paulusma. A Reconfigurations Analogue of Brooks' Theorem and Its Consequences. Journal of Graph Theory, 83(4):340-358, 2016.
19. Philippe Galinier, Michel Habib, and Christophe Paul. Chordal graphs and their clique graphs. In Manfred Nagl, editor, Graph-Theoretic Concepts in Computer Science, pages 358-371, Berlin, Heidelberg, 1995. Springer Berlin Heidelberg.
20. Takehiro Ito, Naonori Kakimura, Naoyuki Kamiyama, Yusuke Kobayashi, and Yoshio Okamoto. Reconfiguration of Maximum-Weight b-Matchings in a Graph. In Computing and Combinatorics - 23rd International Conference, COCOON 2017, Hong Kong, China, August 3-5, 2017, Proceedings, pages 287-296, 2017.
21. Daniel Lokshtanov and Amer E. Mouawad. The complexity of independent set reconfiguration on bipartite graphs. In Proceedings of the Symposium on Discrete Algorithms, SODA 2018, pages 185-195, 2018.
22. Bojan Mohar and Jesús Salas. On the non-ergodicity of the Swendsen-Wang-Kotecký algorithm on the Kagomé lattice. Journal of Statistical Mechanics: Theory and Experiment, 2010(05):P05016, 2010.
23. N. Nishimura. Introduction to reconfiguration. preprint, 2017.
24. A. Suzuki, A. Mouawad, and N. Nishimura. Reconfiguration of Dominating Sets. CoRR, 1401.5714, 2014. URL: http://arxiv.org/abs/1401.5714.
25. J. van den Heuvel. The Complexity of change, page 409. Part of London Mathematical Society Lecture Note Series. Cambridge University Press, S. R. Blackburn, S. Gerke, and M. Wildon edition, 2013.
X

Feedback for Dagstuhl Publishing