Abstract
A clique-transversal set T of a graph G is a set of vertices of G such that T meets all maximal cliques of G. The clique-transversal number, denoted τ c ( G), is the minimum cardinality of a clique-transversal set. Let n be the number of vertices of G. We study classes of graphs G for which n 2 is an upper bound for τ c ( G). Assuming that G has no isolated vertices it is shown that (i) τ c(G)⩽ n 2 for all connected line graphs with the exception of odd cycles, and (ii) τ c(G)⩽ n 2 for all complments of line graphs with the exeption of five small graphs. In addition, a closely related question is studied: call G weakly 2- colorable if its vertices can be colored with 2 colors such that G has no monochromatic maximal clique of size ⩾2. It is proved that a connected line graph G = L( H) is weakly 2-colorable iff H has a 2-coloring of its edges without monochromatic triangles and H is not an odd cycle. Moreover it is shown that complements of line graphs are weakly 2-colorable, with the exception of nine small graphs.
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