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

- Journal title : Bulletin of the Korean Mathematical Society
- Volume 52, Issue 5, 2015, pp.1423-1431
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/BKMS.2015.52.5.1423

Title & Authors

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

KANG, LIYING; SHAN, ERFANG;

KANG, LIYING; SHAN, ERFANG;

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.

Keywords

matching;clique-transversal set;clique-transversal number;cubic graph;line graph;

Language

English

References

1.

T. Andreae, On the clique-transversal number of chordal graphs, Discrete Math. 191 (1998), no. 1-3, 3-11.

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.

3.

S. Aparna Lakshmanan and A. Vijayakumar, The (t)-property of some classes of graphs, Discrete Math. 309 (2009), no. 1, 259-263.

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.

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.

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.

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.

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.

13.

W. T. Tutte, Connectivity in Graphs, University of Toronto Press, Toronto, 1966.