Abstract
A k-star colouring of a graph G is a function \(f:V(G)\rightarrow \{0,1,\dots ,k-1\}\) such that \(f(u)\ne f(v)\) for every edge uv of G, and G does not contain a 4-vertex path bicoloured by f as a subgraph. For \(k\in \mathbb {N}\), the problem k-Star Colourability takes a graph G as input and asks whether G is k-star colourable. By the construction of Coleman and Moré (SIAM J. Numer. Anal., 1983), for all \(k\ge 3\), k-Star Colourability is NP-complete for graphs of maximum degree \(d=k(k-1+\lceil \sqrt{k} \rceil )\). For \(k=4\) and \(k=5\), the maximum degree in this NP-completeness result is \(d=20\) and \(d=35\) respectively. We reduce the maximum degree to \(d=4\) in both cases: i.e., 4-Star Colourability and 5-Star Colourability are NP-complete for graphs of maximum degree four. We also show that for all \(k\ge 3\) and \(d<k\), the time complexity of k-Star Colourability is the same for graphs of maximum degree d and d-regular graphs (i.e., the problem is either in P for both classes or NP-complete for both classes).
M. A. Shalu—Supported by SERB(DST), MATRICS scheme MTR/2018/000086.
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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). https://doi.org/10.37236/1779
Bok, J., Jedlic̆ková, N., Martin, B., Paulusma, D., Smith, S.: Acyclic, star and injective colouring: a complexity picture for H-free graphs. In: Grandoni, F., Herman, G., Sanders, P. (eds.) 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), vol. 173, pp. 22:1–22:22. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Dagstuhl, Germany (2020). https://doi.org/10.4230/LIPIcs.ESA.2020.22
Coleman, T.F., Moré, J.J.: Estimation of sparse Jacobian matrices and graph coloring problems. SIAM J. Numer. Anal. 20(1), 187–209 (1983)
Emden-Weinert, T., Hougardy, S., Kreuter, B.: Uniquely colourable graphs and hardness of colouring graphs of large girth. Comb. Probab. Comput. 7(4), 375–386 (1998). https://doi.org/10.1017/S0963548398003678
Gebremedhin, A.H., Tarafdar, A., Manne, F., Pothen, A.: New acyclic and star coloring algorithms with application to computing Hessians. SIAM J. Sci. Comput. 29(3), 1042–1072 (2007). https://doi.org/10.1137/050639879
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
Grünbaum, B.: Acyclic colorings of planar graphs. Israel J. Math. 14, 390–408 (1973). https://doi.org/10.1007/BF02764716
mikero (https://cstheory.stackexchange.com/users/149/mikero): Parameterized complexity from P to NP-hard and back again. Theoretical Computer Science Stack Exchange. https://cstheory.stackexchange.com/q/3473, (version: 13 April 2017)
Lei, H., Shi, Y., Song, Z.X.: Star chromatic index of subcubic multigraphs. J. Graph Theory 88(4), 566–576 (2018). https://doi.org/10.1002/jgt.22230
Lyons, A.: Acyclic and star colorings of cographs. Discret. Appl. Math. 159(16), 1842–1850 (2011). https://doi.org/10.1016/j.dam.2011.04.011
Molloy, M., Reed, B.: Colouring graphs when the number of colours is almost the maximum degree. J. Comb. Theory Ser. B 109, 134–195 (2014). https://doi.org/10.1016/j.jctb.2014.06.004
Omoomi, B., Roshanbin, E., Dastjerdi, M.V.: A polynomial time algorithm to find the star chromatic index of trees. Electron. J. Comb. 28(1) (2021). https://doi.org/10.37236/9202. Article No. 16
Shalu, M.A., Antony, C.: The complexity of restricted star colouring. Discret. Appl. Math. (2021, in press). https://doi.org/10.1016/j.dam.2021.05.015. Available online: 31 May 2021
West, D.B.: Introduction to graph theory, 2nd edn. Prentice Hall, Upper Saddle River (2001)
Xie, D., Xiao, H., Zhao, Z.: Star coloring of cubic graphs. Inf. Process. Lett. 114(12), 689–691 (2014). https://doi.org/10.1016/j.ipl.2014.05.013
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The first author is supported by SERB(DST), MATRICS scheme MTR/2018/000086. We thank Kirubakaran V. K. and three anonymous referees for their suggestions.
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Shalu, M.A., Antony, C. (2022). The Complexity of Star Colouring in Bounded Degree Graphs and Regular Graphs. In: Balachandran, N., Inkulu, R. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2022. Lecture Notes in Computer Science(), vol 13179. Springer, Cham. https://doi.org/10.1007/978-3-030-95018-7_7
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