On the complexity of bicoloring clique hypergraphs of graphs

Given a graph G, its clique hypergraph C(G) has the same set of vertices as G and the hyperedges correspond to the (inclusionwise) maximal cliques of G. We consider the question of bicolorability of C(G), i.e., whether the vertices of G can be colored with two colors so that no maximal clique is monochromatic. Our two main results say that deciding the bicolorability of C(G) is NP-hard for perfect graphs (and even for those with clique number 3), but solvable in polynomial time for planar graphs.  2002 Elsevier Science (USA). All rights reserved. ✩ The Extended Abstract of this paper was presented at SODA’00 [Proceedings of 11th Annual ACM symposium on Discrete Algorithms, San Francisco, January 2000, ACM and SIAM, 2000, pp. 40–41]. * Corresponding author. E-mail addresses: [email protected] (J. Kratochvíl), [email protected] (Z. Tuza). 1 This author acknowledges further partial support of Czech research grants GAUK 194/1996 and 158/1999, and KONTAKT ME 338/1999 of the Czech Republic. 2 Research supported in part by the Hungarian Scientific Research Fund, grants OTKA T-026575 and T-032969, and Czech Grant GAČR 201/99/0242 (DIMATIA). 3 Project LN00A056 supported by The Ministry of Education. 0196-6774/02/$ – see front matter  2002 Elsevier Science (USA). All rights reserved. PII: S0196-6774(02) 00 22 13 J. Kratochvíl, Z. Tuza / Journal of Algorithms 45 (2002) 40–54 41