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http://dx.doi.org/10.4134/BKMS.2015.52.5.1423

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)
Publication Information
Bulletin of the Korean Mathematical Society / v.52, no.5, 2015 , pp. 1423-1431 More about this Journal
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;
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  • Reference
1 T. Andreae, On the clique-transversal number of chordal graphs, Discrete Math. 191 (1998), no. 1-3, 3-11.   DOI   ScienceOn
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.   DOI   ScienceOn
3 S. Aparna Lakshmanan and A. Vijayakumar, The (t)-property of some classes of graphs, Discrete Math. 309 (2009), no. 1, 259-263.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI
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.   DOI   ScienceOn