{"title":"Notes on Fractional k-Covered Graphs","authors":"Sizhong Zhou, Yang Xu","volume":43,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":971,"pagesEnd":974,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/15704","abstract":"A graph G is fractional k-covered if for each edge e of\r\nG, there exists a fractional k-factor h, such that h(e) = 1. If k = 2,\r\nthen a fractional k-covered graph is called a fractional 2-covered\r\ngraph. The binding number bind(G) is defined as follows,\r\nbind(G) = min{|NG(X)|\r\n|X|\r\n: \u251c\u00ff \u0002= X \u00d4\u00e8\u00e5 V (G),NG(X) \u0002= V (G)}.\r\nIn this paper, it is proved that G is fractional 2-covered if \u03b4(G) \u2265 4\r\nand bind(G) > 5\r\n3 .","references":"[1] D.R. Woodall, The binding number of a graph and its Anderson number,\r\nJ.Combin. Theory ser. B 15(1973), 225-255.\r\n[2] J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, London,\r\nThe Macmillan Press, 1976.\r\n[3] Edward R. Schinerman and D.H. Ullman, Fractional Graph Theory, John\r\nWiley and Son. Inc. New York, 1997.\r\n[4] S. Zhou, Independence number, connectivity and (a, b, k)-critical graphs,\r\nDiscrete Mathematics 309(12)(2009), 4144-4148.\r\n[5] S. Zhou, A new sufficient condition for graphs to be (g, f, n)-critical\r\ngraphs, Canadian Mathematical Bulletin, to appear.\r\n[6] S. Zhou, A sufficient condition for a graph to be an (a, b, k)-critical\r\ngraph, International Journal of Computer Mathematics, to appear.\r\n[7] S. Zhou and Y. Xu, Neighborhoods of independent sets for (a, b, k)-\r\nCritical Graphs, Bulletin of the Australian Mathematical Society\r\n77(2)(2008), 277-283.\r\n[8] G. Liu and L. Zhang, Fractional (g, f)-factors of graphs, Acta Math.\r\nScientia (Ser. B) 21(4)(2001), 541-545.\r\n[9] S. Zhou and Q. Shen, On fractional (f, n)-critical graphs, Information\r\nProcessing Letters 109(14)(2009), 811-815.\r\n[10] S. Zhou, Some results on fractional k-factors, Indian Journal of Pure\r\nand Applied Mathematics 40(2)(2009), 113-121.\r\n[11] S. Zhou and H. Liu, Neighborhood conditions and fractional k-factors,\r\nBulletin of the Malaysian Mathematical Sciences Society 32(1)(2009),\r\n37-45.\r\n[12] S. Zhou and C. Shang, Some sufficient conditions with fractional\r\n(g, f)-factors in graphs, Chinese Journal of Engineering Mathematics\r\n24(2)(2007), 329-333.\r\n[13] Z. Li, G. Yan and X. Zhang, On fractional f-covered graphs, OR\r\nTrasactions (in Chinese) 6(4)(2002), 65-68.\r\n[14] Z. Li, G. Yan and X. Zhang, Isolated toughness and fractional k-covered\r\ngraphs, Acta mathematicae Applicatae Sinica 27(4)(2004), 593-598.\r\n[15] R. R. Anstee, An Algorithmic Proof Tutte-s f-Factor Theorem, J.\r\nAlgorithms 6(1985), 112-131.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 43, 2010"}