Abstract
A function \(c:V(G)\rightarrow \{1,2,\ldots ,k\}\) is a k-colouring of a graph G if \(c(u)\ne c(v)\) whenever u and v are adjacent. If any two colour classes induce the disjoint union of vertices and edges, then c is called injective. Injective colourings are also known as L(1, 1)-labellings and distance 2-colourings. The corresponding decision problem is denoted Injective Colouring. A graph is H-free if it does not contain H as an induced subgraph. We prove a dichotomy for Injective Colouring for graphs with bounded independence number. Then, by combining known with further new results, we determine the complexity of Injective Colouring on H-free graphs for every H except for one missing case.
The research in this paper was supported by GAUK 1580119, SVV–2020–260578 and the Leverhulme Trust (RPG-2016-258).
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Bodlaender, H.L., Kloks, T., Tan, R.B., van Leeuwen, J.: Approximations for lambda-colorings of graphs. Comput. J. 47, 193–204 (2004)
Bok, J., Jedlic̆ková, N., Martin, B., Paulusma, D., Smith, S.: Acyclic colouring, star colouring and injective colouring for \({H}\)-free graphs. In: Proceedings ESA 2020, LIPIcs, vol. 173, pp. 22:1–22:22 (2020)
Calamoneri, T.: The \({L}(h, k)\)-labelling problem: an updated survey and annotated bibliography. Comput. J. 54, 1344–1371 (2011)
Chudnovsky, M., Huang, S., Spirkl, S., Zhong, M.: List-three-coloring graphs with no induced \({P}_6+r{P}_3\). CoRR, abs/1806.11196 (2018)
Golovach, P.A., Johnson, M., Paulusma, D., Song, J.: A survey on the computational complexity of colouring graphs with forbidden subgraphs. J. Graph Theory 84, 331–363 (2017)
Hahn, G., Kratochvíl, J., Širáň, J., Sotteau, D.: On the injective chromatic number of graphs. Discret. Math. 256, 179–192 (2002)
Heggernes, P., Telle, J.A.: Partitioning graphs into generalized dominating sets. Nordic J. Comput. 5, 128–142 (1998)
Hell, P., Raspaud, A., Stacho, J.: On injective colourings of chordal graphs. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds.) LATIN 2008. LNCS, vol. 4957, pp. 520–530. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78773-0_45
Jansen, K.: Complexity Results for the Optimum Cost Chromatic Partition Problem, Universität Trier, Mathematik/Informatik, Forschungsbericht, pp. 96–41 (1996)
Jin, J., Baogang, X., Zhang, X.: On the complexity of injective colorings and its generalizations. Theoret. Comput. Sci. 491, 119–126 (2013)
Klimošová, T., Malík, J., Masařík, T., Novotná, J., Paulusma, D., Slívová, V.: Colouring \(({P}_r+{P}_s)\)-free graphs. In: Proceedings ISAAC 2018, LIPIcs, pp. 123:5:1–5:13 (2018)
Král’, D., Kratochvíl, J., Tuza, Z., Woeginger, G.J.: Complexity of coloring graphs without forbidden induced subgraphs. In: Brandstädt, A., Le, V.B. (eds.) WG 2001. LNCS, vol. 2204, pp. 254–262. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45477-2_23
Lloyd, E.L., Ramanathan, S.: On the complexity of distance-\(2\) coloring. Proc. ICCI 1992, 71–74 (1992)
Mahdian, M.: On the computational complexity of strong edge coloring. Discret. Appl. Math. 118, 239–248 (2002)
Sen, A., Huson, M.L.: A new model for scheduling packet radio networks. Wireless Netw. 3, 71–82 (1997)
Zhou, X., Kanari, Y., Nishizeki, T.: Generalized vertex-coloring of partial \(k\)-trees. IEICE Trans. Fundamentals Electron. Commun. Comput. Sci. E83-A, 671–678 (2000)
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Bok, J., Jedličková, N., Martin, B., Paulusma, D., Smith, S. (2021). Injective Colouring for H-Free Graphs. In: Santhanam, R., Musatov, D. (eds) Computer Science – Theory and Applications. CSR 2021. Lecture Notes in Computer Science(), vol 12730. Springer, Cham. https://doi.org/10.1007/978-3-030-79416-3_2
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