Abstract
A rainbow colouring of a connected graph is a colouring of the edges of the graph, such that every pair of vertices is connected by at least one path in which no two edges are coloured the same. Such a colouring using minimum possible number of colours is called an optimal rainbow colouring, and the minimum number of colours required is called the rainbow connection number of the graph. A Chordal Graph is a graph in which every cycle of length more than 3 has a chord. A Split Graph is a chordal graph whose vertices can be partitioned into a clique and an independent set. A threshold graph is a split graph in which the neighbourhoods of the independent set vertices form a linear order under set inclusion. In this article, we show the following:
-
1
The problem of deciding whether a graph can be rainbow coloured using 3 colours remains NP-complete even when restricted to the class of split graphs. However, any split graph can be rainbow coloured in linear time using at most one more colour than the optimum.
-
2
For every integer k ≥ 3, the problem of deciding whether a graph can be rainbow coloured using k colours remains NP-complete even when restricted to the class of chordal graphs.
-
3
For every positive integer k, threshold graphs with rainbow connection number k can be characterised based on their degree sequence alone. Further, we can optimally rainbow colour a threshold graph in linear time.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ananth, P., Nasre, M., Sarpatwar, K.K.: Rainbow Connectivity: Hardness and Tractability. In: IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011), vol. 13, pp. 241–251 (2011)
Basavaraju, M., Chandran, L.S., Rajendraprasad, D., Ramaswamy, A.: Rainbow connection number and radius. Preprint arXiv:1011.0620v1 (math.CO) (2010)
Basavaraju, M., Chandran, L.S., Rajendraprasad, D., Ramaswamy, A.: Rainbow connection number of graph power and graph products. Preprint arXiv:1104.4190v2 (math.CO) (2011)
Caro, Y., Lev, A., Roditty, Y., Tuza, Z., Yuster, R.: On rainbow connection. Electron. J. Combin. 15(1), Research paper 57, 13 (2008)
Chakraborty, S., Fischer, E., Matsliah, A., Yuster, R.: Hardness and algorithms for rainbow connection. J. Comb. Optim. 21(3), 330–347 (2011)
Chandran, L.S., Das, A., Rajendraprasad, D., Varma, N.M.: Rainbow connection number and connected dominating sets. Journal of Graph Theory (2011)
Chandran, L.S., Rajendraprasad, D.: Rainbow colouring of split and threshold graphs. Preprint arXiv:1205.1670v1 (cs.DM) (2012)
Chartrand, G., Zhang, P.: Chromatic Graph Theory. Chapman & Hall (2008)
Chartrand, G., Johns, G.L., McKeon, K.A., Zhang, P.: Rainbow connection in graphs. Math. Bohem. 133(1), 85–98 (2008)
Cover, T.M., Thomas, J.A.: Data Compression, pp. 103–158. John Wiley & Sons, Inc. (2005)
Frieze, A., Tsourakakis, C.: Rainbow connectivity of g (n, p) at the connectivity threshold. Preprint arXiv:1201.4603 (2012)
Hammer, P.L., Simeone, B.: The splittance of a graph. Combinatorica 1(3), 275–284 (1981)
He, J., Liang, H.: On rainbow k-connectivity of random graphs. Preprint arXiv:1012.1942v1 (math.CO) (2010)
Holyer, I.: The NP-completeness of edge-coloring. SIAM Journal on Computing 10(4), 718–720 (1981)
Khot, S.: Hardness results for coloring 3-colorable 3-uniform hypergraphs. In: Proceedings of The 43rd Annual IEEE Symposium on Foundations of Computer Science, pp. 23–32. IEEE (2002)
Kraft, L.: A device for quanitizing, grouping and coding amplitude modulated pulses. Master’s thesis. Electrical Engineering Department, Massachusetts Institute of Technology (1949)
Krivelevich, M., Yuster, R.: The rainbow connection of a graph is (at most) reciprocal to its minimum degree. J. Graph Theory 63(3), 185–191 (2010)
Li, S., Li, X.: Note on the complexity of determining the rainbow connectedness for bipartite graphs. Preprint arXiv:1109.5534 (2011)
Li, X., Sun, Y.: Rainbow Connections of Graphs. Springerbriefs in Mathematics. Springer (2012)
Li, X., Sun, Y.: Rainbow connections of graphs – a survey. Preprint arXiv:1101.5747v2 (math.CO) (2011)
Li, X., Sun, Y.: Upper bounds for the rainbow connection numbers of line graphs. Graphs and Combinatorics, 1–13 (2011), doi:10.1007/s00373-011-1034-1
Misra, J., Gries, D.: A constructive proof of vizing’s theorem. Information Processing Letters 41(3), 131–133 (1992)
Schiermeyer, I.: Rainbow Connection in Graphs with Minimum Degree Three. In: Fiala, J., KratochvÃl, J., Miller, M. (eds.) IWOCA 2009. LNCS, vol. 5874, pp. 432–437. Springer, Heidelberg (2009)
Seward, H.H.: Information sorting in the application of electronic digital computers to business operations. Master’s thesis, Digital Computer Laboratory, Massachusetts Institute of Technology (1954)
Shang, Y.: A sharp threshold for rainbow connection of random bipartite graphs. Int. J. Appl. Math. 24(1), 149–153 (2011)
Wigderson, A.: The complexity of graph connectivity. In: Mathematical Foundations of Computer Science, pp. 112–132 (1992)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chandran, L.S., Rajendraprasad, D. (2012). Rainbow Colouring of Split and Threshold Graphs. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds) Computing and Combinatorics. COCOON 2012. Lecture Notes in Computer Science, vol 7434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32241-9_16
Download citation
DOI: https://doi.org/10.1007/978-3-642-32241-9_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32240-2
Online ISBN: 978-3-642-32241-9
eBook Packages: Computer ScienceComputer Science (R0)