Advertisement

LATIN 2018: LATIN 2018: Theoretical Informatics pp 625-639

# Algorithms and Bounds for Very Strong Rainbow Coloring

• L. Sunil Chandran
• Anita Das
• Davis Issac
• Erik Jan van Leeuwen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)

## Abstract

A well-studied coloring problem is to assign colors to the edges of a graph G so that, for every pair of vertices, all edges of at least one shortest path between them receive different colors. The minimum number of colors necessary in such a coloring is the strong rainbow connection number ($$\mathbf {src}(G)$$) of the graph. When proving upper bounds on $$\mathbf {src}(G)$$, it is natural to prove that a coloring exists where, for every shortest path between every pair of vertices in the graph, all edges of the path receive different colors. Therefore, we introduce and formally define this more restricted edge coloring number, which we call very strong rainbow connection number ($$\mathbf {vsrc}(G)$$).

In this paper, we give upper bounds on $$\mathbf {vsrc}(G)$$ for several graph classes, some of which are tight. These immediately imply new upper bounds on $$\mathbf {src}(G)$$ for these classes, showing that the study of $$\mathbf {vsrc}(G)$$ enables meaningful progress on bounding $$\mathbf {src}(G)$$. Then we study the complexity of the problem to compute $$\mathbf {vsrc}(G)$$, particularly for graphs of bounded treewidth, and show this is an interesting problem in its own right. We prove that $$\mathbf {vsrc}(G)$$ can be computed in polynomial time on cactus graphs; in contrast, this question is still open for $$\mathbf {src}(G)$$. We also observe that deciding whether $$\mathbf {vsrc}(G) = k$$ is fixed-parameter tractable in k and the treewidth of G. Finally, on general graphs, we prove that there is no polynomial-time algorithm to decide whether $$\mathbf {vsrc}(G) \le 3$$ nor to approximate $$\mathbf {vsrc}(G)$$ within a factor $$n^{1-\varepsilon }$$, unless $$\text {P}=\text {NP}$$.

## References

1. 1.
Ananth, P., Nasre, M., Sarpatwar, K.K.: Rainbow connectivity: Hardness and tractability. In: Chakraborty, S., Kumar, A. (eds.) Proceedings of FSTTCS 2011. LIPIcs, vol. 13, pp. 241–251. Schloss Dagstuhl (2011)Google Scholar
2. 2.
Basavaraju, M., Chandran, L.S., Rajendraprasad, D., Ramaswamy, A.: Rainbow connection number and radius. Graphs Comb. 30(2), 275–285 (2014)
3. 3.
Björklund, A., Husfeldt, T., Koivisto, M.: Set partitioning via inclusion-exclusion. SIAM J. Comput. 39, 546–563 (2009)
4. 4.
Chakraborty, S., Fischer, E., Matsliah, A., Yuster, R.: Hardness and algorithms for rainbow connection. J. Comb. Optim. 21(3), 330–347 (2011)
5. 5.
Chandran, L.S., Das, A., Rajendraprasad, D., Varma, N.M.: Rainbow connection number and connected dominating sets. J. Graph Theory 71(2), 206–218 (2012)
6. 6.
Chandran, L.S., Rajendraprasad, D.: Rainbow colouring of split and threshold graphs. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds.) COCOON 2012. LNCS, vol. 7434, pp. 181–192. Springer, Heidelberg (2012).
7. 7.
Chandran, L.S., Rajendraprasad, D.: Inapproximability of rainbow colouring. In: Seth, A., Vishnoi, N.K. (eds.) Proceedings of FSTTCS 2013. LIPIcs, vol. 24, pp. 153–162. Schloss Dagstuhl (2013)Google Scholar
8. 8.
Chartrand, G., Johns, G.L., McKeon, K.A., Zhang, P.: Rainbow connection in graphs. Math. Bohem. 133(1), 85–98 (2008)
9. 9.
Eiben, E., Ganian, R., Lauri, J.: On the complexity of rainbow coloring problems. Discret. Appl. Math. (2016, in press).
10. 10.
Erdős, P., Goodman, A.W., Pósa, L.: The representation of a graph by set intersections. Canad. J. Math 18(106–112), 86 (1966)
11. 11.
Gary, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)Google Scholar
12. 12.
Gavril, F.: The intersection graphs of subtrees in trees are exactly the chordal graphs. JCTB 16(1), 47–56 (1974)
13. 13.
Kammer, F., Tholey, T.: Approximation algorithms for intersection graphs. Algorithmica 68(2), 312–336 (2014)
14. 14.
Keranen, M., Lauri, J.: Computing minimum rainbow and strong rainbow colorings of block graphs. arXiv preprint arXiv:1405.6893 (2014)
15. 15.
Kowalik, Ł., Lauri, J., Socala, A.: On the fine-grained complexity of rainbow coloring. In: Sankowski, P., Zaroliagis, C.D. (eds.) Proceedings of ESA 2016. LIPIcs, vol. 57, pp. 58:1–58:16. Schloss Dagstuhl (2016)Google Scholar
16. 16.
Lauri, J.: Chasing the Rainbow Connection: Hardness, Algorithms, and Bounds, vol. 1428. Tampere University of Technology Publication, Tampere (2016)Google Scholar
17. 17.
Li, X., Shi, Y., Sun, Y.: Rainbow connections of graphs: a survey. Graphs Comb. 29(1), 1–38 (2013)
18. 18.
Li, X., Sun, Y.: Rainbow Connections of Graphs. Springer Science & Business Media, Boston (2012).
19. 19.
Li, X., Sun, Y.: An updated survey on rainbow connections of graphs - a dynamic survey. Theory Appl. Graphs 0, 3 (2017)Google Scholar
20. 20.
Roberts, F.S.: Applications of edge coverings by cliques. Discret. Appl. Math. 10(1), 93–109 (1985)
21. 21.
Robertson, N., Seymour, P.D.: Graph minors. II. Algorithmic aspects of tree-width. JCTB 7, 309–322 (1986)
22. 22.
Uchizawa, K., Aoki, T., Ito, T., Suzuki, A., Zhou, X.: On the rainbow connectivity of graphs: complexity and FPT algorithms. Algorithmica 67(2), 161–179 (2013)
23. 23.
Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. In: Proceedings of STOC 2006, pp. 681–690. ACM (2006)Google Scholar

## Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

## Authors and Affiliations

• L. Sunil Chandran
• 1
• Anita Das
• 2
• Davis Issac
• 3
• Erik Jan van Leeuwen
• 4
1. 1.Department of Computer Science and AutomationIndian Institute of ScienceBangaloreIndia
2. 2.Infosys Ltd.BangaloreIndia
3. 3.MPI für Informatik, Saarland Informatics CampusSaarbrückenGermany
4. 4.Department of Information and Computing SciencesUtrecht UniversityUtrechtThe Netherlands