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BINDING NUMBER AND MINIMUM DEGREE FOR FRACTIONAL (k,m)-DELETED GRAPHS

  • SIZHONG ZHOU (a1), QIUXIANG BIAN (a2) and LAN XU (a3)
Abstract
Abstract

Let G be a graph of order n, and let k≥1 be an integer. Let h:E(G)→[0,1] be a function. If ∑ exh(e)=k holds for any xV (G), then we call G[Fh] a fractional k-factor of G with indicator function h where Fh ={eE(G):h(e)>0}. A graph G is called a fractional (k,m) -deleted graph if for every eE(H) , there exists a fractional k-factor G[Fh ] of G with indicator function h such that h(e)=0 , where H is any subgraph of G with m edges. The minimum degree of a vertex in G is denoted by δ(G) . For XV (G), NG(X)=⋃ xXNG(x) . The binding number of G is defined by In this paper, it is proved that if then G is a fractional (k,m) -deleted graph. Furthermore, it is shown that this result is best possible in some sense.

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Corresponding author
For correspondence; e-mail: zsz_cumt@163.com
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This research was supported by Natural Science Foundation of the Higher Education Institutions of Jiangsu Province (10KJB110003) and Jiangsu University of Science and Technology (2010SL101J, 2009SL154J), and was sponsored by Qing Lan Project of Jiangsu Province.

Footnotes
References
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[1]Bondy J. A. and Murty U. S. R., Graph Theory with Applications (Macmillan Press, London, 1976).
[2]Cai J., Liu G. and Hou J., ‘The stability number and connected [k,k+1]-factor in graphs’, Appl. Math. Lett. 22(6) (2009), 927931.
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[4]Liu H. and Liu G., ‘Neighbor set for the existence of (g,f,n)-critical graphs’, Bull. Malays. Math. Sci. Soc. (2) 34(1) (2011), 3949.
[5]Liu G. and Zhang L., ‘Toughness and the existence of fractional k-factors of graphs’, Discrete Math. 308 (2008), 17411748.
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[7]Yu J., Liu G., Ma M. and Cao B., ‘A degree condition for graphs to have fractional factors’, Adv. Math. (China) 35(5) (2006), 621628.
[8]Zhou S., ‘Binding number and minimum degree for the existence of fractional k-factors with prescribed properties’, Util. Math., to appear.
[9]Zhou S., ‘Independence number, connectivity and (a,b,k)-critical graphs’, Discrete Math. 309(12) (2009), 41444148.
[10]Zhou S., ‘A neighborhood condition for graphs to be fractional (k,m)-deleted graphs’, Glasg. Math. J. 52(1) (2010), 3340.
[11]Zhou S., ‘Some new sufficient conditions for graphs to have fractional k-factors’, Int. J. Comput. Math. 88(3) (2011), 484490.
[12]Zhou S., ‘A sufficient condition for graphs to be fractional (k,m)-deleted graphs’, Appl. Math. Lett. 24(9) (2011), 15331538.
[13]Zhou S. and Jiang J., ‘Toughness and (a,b,k)-critical graphs’, Inform. Process. Lett. 111(9) (2011), 403407.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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