# CLIQUE-TRANSVERSAL SETS IN LINE GRAPHS OF CUBIC GRAPHS AND TRIANGLE-FREE GRAPHS

• KANG, LIYING (DEPARTMENT OF MATHEMATICS SHANGHAI UNIVERSITY) ;
• SHAN, ERFANG (DEPARTMENT OF MATHEMATICS SHANGHAI UNIVERSITY, SCHOOL OF MANAGEMENT SHANGHAI UNIVERSITY)
• Received : 2013.11.06
• Published : 2015.09.30

#### Abstract

A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G. The 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.

#### Acknowledgement

Supported by : National Nature Science Foundation of China

#### References

1. T. Andreae, On the clique-transversal number of chordal graphs, Discrete Math. 191 (1998), no. 1-3, 3-11. https://doi.org/10.1016/S0012-365X(98)00087-9
2. T. Andreae, M. Schughart, and Zs. Tuza, Clique-transversal sets of line graphs and complements of line graphs, Discrete Math. 88 (1991), no. 1, 11-20. https://doi.org/10.1016/0012-365X(91)90055-7
3. S. Aparna Lakshmanan and A. Vijayakumar, The (t)-property of some classes of graphs, Discrete Math. 309 (2009), no. 1, 259-263. https://doi.org/10.1016/j.disc.2007.12.057
4. G. Bacso and Zs. Tuza, Clique-transversal sets and weak 2-colorings in graphs of small maximum degree, Discrete Math. Theor. Comput. Sci. 11 (2009), no. 2, 15-24.
5. C. Berge, Hypergraphs, Amsterdam: North-Holland, 1989.
6. T. Biedl, E. D. Demaine, C. A. Duncan, R. Fleischer, and S. G. Kobourov, Tight bounds on maximal and maximum matchings, Discrete Math. 285 (2004), no. 1-3, 7-15. https://doi.org/10.1016/j.disc.2004.05.003
7. J. A. Bondy and U. S. R. Murty, Graph Theory, Springer, 2008.
8. P. Erdos, T. Gallai, and Zs. Tuza, Covering the cliques of a graph with vertices, Discrete Math. 108 (1992), no. 1-3, 279-289. https://doi.org/10.1016/0012-365X(92)90681-5
9. T. Gallai, Uber extreme Punkt-und Kantenmengen, Ann. Univ. Sci. Budapest. Eotvos Sect. Math. 2 (1959), 133-138.
10. A. M. Hobbs and E. Schmeichel, On the maximum number of independent edges in cubic graphs, Discrete Math. 42 (1982), no. 2-3, 317-320. https://doi.org/10.1016/0012-365X(82)90227-8
11. E. F. Shan, T. C. E. Cheng, and L. Y. Kang, Bounds on the clique-transversal number of regular graphs, Sci. China Ser. A 51 (2008), no. 5, 851-863. https://doi.org/10.1007/s11425-007-0157-6
12. O. Suil and D. B. West, Balloons, cut-edges, matchings, and total domination in regular graphs of odd degree, J. Graph Theory 64 (2010), no. 2, 116-131. https://doi.org/10.1002/jgt.20443
13. W. T. Tutte, Connectivity in Graphs, University of Toronto Press, Toronto, 1966.
14. Zs. Tuza, Covering all cliques of a graph, Discrete Math. 86 (1990), no. 1-3, 117-126. https://doi.org/10.1016/0012-365X(90)90354-K