Abstract
The clique-chromatic number of a graph G = (V,E) denoted by χ c (G) is the smallest integer k such that there exists a partition of the vertex set of G into k subsets with the property that no maximal clique of G is contained in any of the subsets. Such a partition is called a k-clique-colouring of G. Recently Marx proved that deciding whether a graph admits a k-clique-colouring is \(\Sigma^p_2\)-complete for every fixed k ≥ 2. Our main results are an O *(2n) time inclusion-exclusion algorithm to compute χ c (G) exactly, and a branching algorithm to decide whether a graph of bounded clique-size admits a 2-clique-colouring which runs in time O *(λ n) for some λ < 2.
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References
Bacsó, G., Gravier, S., Gyárfás, A., Preissmann, M., Sebö, A.: Coloring the maximal cliques of graphs. SIAM Journal on Discrete Mathematics 17(3), 361–376 (2004)
Berge, C., Duchet, P.: Strongly perfect graphs. Annals of Discrete Mathematics 21, 57–61 (1984)
Berge, C.: Hypergraphs: combinatorics of finite sets. North holland (1984)
Björklund, A., Husfeldt, T., Koivisto, M.: Set Partitioning via Inclusion-Exclusion. SIAM J. Comput. 39(2), 546–563 (2009)
Défossez, D.: Clique-coloring some classes of odd-hole-free graphs. Journal of Graph Theory 53(3), 233–249 (2006)
Duffus, D., Sands, B., Sauer, N., Woodrow, R.: Two-colouring all two-element maximal antichains. Journal of Combinatorial Theory, Series A 57(1), 109–116 (1991)
Fomin, F.V., Kratsch, D.: Exact exponential algorithms. Springer (2011)
Gaspers, S.: Algorithmes exponentiels. Master’s thesis, Université de Metz (June 2005)
Klein, S., Morgana, A.: On clique-colouring of graphs with few P4’s. Journal of the Brazilian Computer Society 18(2), 113–119 (2012)
Kratochvíl, J., Tuza, Z.: On the complexity of bicoloring clique hypergraphs of graphs. Journal of Algorithms 45(1), 40–54 (2002)
Kullmann, O.: New methods for 3-SAT decision and worst-case analysis. Theor. Comput. Sci. 223(1-2), 1–72 (1999)
Leiserson, C.E., Rivest, R.L., Stein, C., Cormen, T.H.: Introduction to algorithms. The MIT press (2001)
Marx, D.: Complexity of clique coloring and related problems. Theoretical Computer Science 412(29), 3487–3500 (2011)
Mohar, B., Skrekovski, R.: The Grötzsch theorem for the hypergraph of maximal cliques. The Electronic Journal of Combinatorics 6(R26), 2 (1999)
Moon, J., Moser, L.: On cliques in graphs. Israel Journal of Mathematics 3(1), 23–28 (1965)
Tsukiyama, S., Ide, M., Ariyoshi, H., Shirakawa, I.: A new algorithm for generating all the maximal independent sets. SIAM Journal on Computing 6(3), 505–517 (1977)
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Cochefert, M., Kratsch, D. (2014). Exact Algorithms to Clique-Colour Graphs. In: Geffert, V., Preneel, B., Rovan, B., Štuller, J., Tjoa, A.M. (eds) SOFSEM 2014: Theory and Practice of Computer Science. SOFSEM 2014. Lecture Notes in Computer Science, vol 8327. Springer, Cham. https://doi.org/10.1007/978-3-319-04298-5_17
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DOI: https://doi.org/10.1007/978-3-319-04298-5_17
Publisher Name: Springer, Cham
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