Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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REMARKS ON NEIGHBORHOODS OF INDEPENDENT SETS AND (a, b, k)-CRITICAL GRAPHS

  • Zhou, Sizhong (School of Mathematics and Physics, Jiangsu University of Science and Technology) ;
  • Sun, Zhiren (School of Mathematics Science, Nanjing Normal University) ;
  • Xu, Lan (Department of Mathematics, Changji University)
  • Received : 2013.01.24
  • Accepted : 2013.04.11
  • Published : 2013.09.30
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Abstract

Let $a$ and $b$ be two even integers with $2{\leq}a<b$, and let k be a nonnegative integer. Let G be a graph of order $n$ with $n{\geq}\frac{(a+b-1)(a+b-2)+bk-2}{b}$. A graph G is called an ($a,b,k$)-critical graph if after deleting any $k$ vertices of G the remaining graph of G has an [$a,b$]-factor. In this paper, it is proved that G is an ($a,b,k$)-critical graph if $${\mid}N_G(X){\mid}&gt;\frac{(a-1)n+{\mid}X{\mid}+bk-2}{a+b-1}$$ for every non-empty independent subset X of V (G), and $${\delta}(G)>\frac{(a-1)n+a+b+bk-3}{a+b-1}$$. Furthermore, it is shown that the result in this paper is best possible in some sense.

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