Abstract
Restricted star colouring is a variant of star colouring introduced to design heuristic algorithms to estimate sparse Hessian matrices. For \( k\in \mathbb {N} \), a \( k \)-restricted star (\( k \)-rs) colouring of a graph \( G \) is a function \( f:V(G)\rightarrow \{0,1,\dots ,k\,-\,1\} \) such that (i) \( f(x)\ne f(y) \) for every edge \( xy \) of \( G \), and (ii) there is no bicoloured 3-vertex path(\( P_3 \)) in \( G \) with the higher colour on its middle vertex. We show that for \( k\ge 3 \), it is NP-complete to decide whether a given planar bipartite graph of maximum degree \( k \) and girth at least six is \( k \)-rs colourable, and thereby answer a problem posed by Shalu and Sandhya (Graphs and Combinatorics 2016). In addition, we design an \( O(n^3) \) algorithm to test whether a chordal graph is 3-rs colourable.
First author is supported by SERB (DST), MATRICS scheme MTR/2018/000086.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Albertson, M.O., Chappell, G.G., Kierstead, H.A., Kündgen, A., Ramamurthi, R.: Coloring with no 2-colored \(P_4\)’s. Electron. J. Comb. 11(1), 26 (2004)
Almeter, J., Demircan, S., Kallmeyer, A., Milans, K.G., Winslow, R.: Graph 2-rankings. Graphs Comb. 35(1), 91–102 (2019). https://doi.org/10.1007/s00373-018-1979-4
Bodlaender, H.L., et al.: Rankings of graphs. SIAM J. Discrete Math. 11(1), 168–181 (1998). https://doi.org/10.1137/S0895480195282550
Bozdağ, D., Çatalyürek, Ü.V., Gebremedhin, A.H., Manne, F., Boman, E.G., Özgüner, F.: Distributed-memory parallel algorithms for distance-2 coloring and related problems in derivative computation. SIAM J. Sci. Comput. 32(4), 2418–2446 (2010). https://doi.org/10.1137/080732158
Curtis, A.R., Powell, M.J., Reid, J.K.: On the estimation of sparse Jacobian matrices. J. Inst. Math. Appl. 13(1), 117–120 (1974). https://doi.org/10.1093/imamat/13.1.117
Dereniowski, D.: Rank coloring of graphs. In: Kubale, M. (ed.) Graph Colorings, pp. 79–93, Chap. 6. American Mathematical Society (2004). https://doi.org/10.1090/conm/352/06
Gebremedhin, A., Nguyen, D., Patwary, M.M.A., Pothen, A.: ColPack: software for graph coloring and related problems in scientific computing. ACM Trans. Math. Softw. (TOMS) 40 (2013). https://doi.org/10.1145/2513109.2513110
Gebremedhin, A.H., Manne, F., Pothen, A.: What color is your Jacobian? Graph coloring for computing derivatives. SIAM Rev. 47(4), 629–705 (2005). https://doi.org/10.1137/S0036144504444711
Karpas, I., Neiman, O., Smorodinsky, S.: On vertex rankings of graphs and its relatives. Discrete Math. 338(8), 1460–1467 (2015). https://doi.org/10.1016/j.disc.2015.03.008
Katchalski, M., McCuaig, W., Seager, S.: Ordered colourings. Discrete Math. 142(1–3), 141–154 (1995). https://doi.org/10.1016/0012-365X(93)E0216-Q
Lyons, A.: Acyclic and star colorings of cographs. Discrete Appl. Math. 159(16), 1842–1850 (2011). https://doi.org/10.1016/j.dam.2011.04.011
Moore, C., Robson, J.M.: Hard tiling problems with simple tiles. Discrete Comput. Geom. 26(4), 573–590 (2001). https://doi.org/10.1007/s00454-001-0047-6
Powell, M., Toint, P.L.: On the estimation of sparse Hessian matrices. SIAM J. Numer. Anal. 16(6), 1060–1074 (1979). https://doi.org/10.1137/0716078
Scheffler, P.: Node ranking and searching on graphs. In: 3rd Twente Workshop on Graphs and Combinatorial Optimization, Memorandum No. 1132 (1993)
Shalu, M.A., Sandhya, T.P.: Star coloring of graphs with girth at least five. Graphs Comb. 32(5), 2121–2134 (2016). https://doi.org/10.1007/s00373-016-1702-2
West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice Hall, Upper Saddle River (2001)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Shalu, M.A., Antony, C. (2020). Complexity of Restricted Variant of Star Colouring. In: Changat, M., Das, S. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2020. Lecture Notes in Computer Science(), vol 12016. Springer, Cham. https://doi.org/10.1007/978-3-030-39219-2_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-39219-2_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-39218-5
Online ISBN: 978-3-030-39219-2
eBook Packages: Computer ScienceComputer Science (R0)