Liying Kang and Erfang Shan Shanghai University, Shanghai University

Abstract : A {\em clique-transversal set} $D$ of a graph $G$ is a set of vertices of $G$ such that $D$ meets all cliques of $G$. The {\em clique-transversal number} is the minimum cardinality of a clique-transversal set in $G$. For every cubic graph with at most two bridges, we first show that it has a perfect matching which contains exactly one edge of each triangle of it; by the result, we determine the exact value of the clique-transversal number of line graph of it. Also, we present a sharp upper bound on the clique-transversal number of line graph of a cubic graph. Furthermore, we prove that the clique-transversal number of line graph of a triangle-free graph is at most the chromatic number of complement of the triangle-free graph.