Chains, antichains, and fibres
Journal of Combinatorial Theory Series A
Clique-transversal sets of line graphs and complements of line graphs
Discrete Mathematics
Covering all cliques of a graph
Discrete Mathematics - Topics on domination
Fibres and ordered set coloring
Journal of Combinatorial Theory Series A
Covering the cliques of a graph with vertices
Discrete Mathematics - Topological, algebraical and combinatorial structures; Froli´k's memorial volume
Algorithmic aspects of neighborhood numbers
SIAM Journal on Discrete Mathematics
Extending matchings in claw-free graphs
Selected papers of the 13th British Combinatorial Conference on British combinatorial conference
On covering all cliques of a chordal graph
Discrete Mathematics
Maximum h-colourable subgraph problem in balanced graphs
Information Processing Letters
On the clique-transversal number of chordal graphs
Discrete Mathematics
Discrete Mathematics
On the complexity of bicoloring clique hypergraphs of graphs
Journal of Algorithms
Coloring the Maximal Cliques of Graphs
SIAM Journal on Discrete Mathematics
Partial characterizations of clique-perfect graphs I: Subclasses of claw-free graphs
Discrete Applied Mathematics
Clique-coloring some classes of odd-hole-free graphs
Journal of Graph Theory
2-List-coloring planar graphs without monochromatic triangles
Journal of Combinatorial Theory Series B
Graph Theory
Complexity of clique-coloring odd-hole-free graphs
Journal of Graph Theory
Distance-hereditary graphs are clique-perfect
Discrete Applied Mathematics
Complexity of clique coloring and related problems
Theoretical Computer Science
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Let G=(V,E) be a graph. A clique-transversal setD is a subset of vertices of G such that D meets all cliques of G, where a clique is defined as a complete subgraph maximal under inclusion and having at least two vertices. The clique-transversal number, denoted by @t"C(G), of G is the cardinality of a minimum clique-transversal set in G. A k-clique-coloring of G is a k-coloring of its vertices such that no clique is monochromatic. All planar graphs have been proved to be 3-clique-colorable by Mohar and Skrekovski [B. Mohar, R. Skrekovski, The Grotzsch theorem for the hypergraph of maximal cliques, Electron. J. Combin. 6 (1999) #R26]. Erdos et al. [P. Erdos, T. Gallai, Zs. Tuza, Covering the cliques of a graph with vertices, Discrete Math. 108 (1992) 279-289] proposed to find sharp estimates on @t"C(G) for planar graphs. In this paper we first show that every outerplanar graph G of order n(=2) has @t"C(G)@?3n/5 and the bound is tight. Secondly, we prove that every claw-free planar graph different from an odd cycle is 2-clique-colorable and we present a polynomial-time algorithm to find the 2-clique-coloring. As a by-product of the result, we obtain a tight upper bound on the clique-transversal number for claw-free planar graphs.