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A SUFFICIENT CONDITION FOR A GRAPH TO BE A FRACTIONAL (f, n)-CRITICAL GRAPH
- Source :
- Glasgow Mathematical Journal. 52:409-415
- Publication Year :
- 2010
- Publisher :
- Cambridge University Press (CUP), 2010.
-
Abstract
- Let a, b and n be non-negative integers such that 1 ≤ a ≤ b, and let G be a graph of order p with $\(p\geq\frac{(a+b-1)(a+b-2)+bn-2}{a}\)$ and f be an integer-valued function defined on V(G) such that a ≤ f(x) ≤ b for all x ∈ V(G). Let h: E(G) → [0, 1] be a function. If ∑e∋xh(e) = f(x) holds for any x ∈ V(G), then we call G[Fh] a fractional f-factor of G with indicator function h, where Fh = {e ∈ E(G): h(e) > 0}. A graph G is called a fractional (f, n)-critical graph if after deleting any n vertices of G the remaining graph of G has a fractional f-factor. In this paper, it is proved that G is a fractional (f, n)-critical graph if $\(|N_G(X)|>\frac{(b-1)p+|X|+bn-1}{a+b-1}\)$ for every non-empty independent subset X of V(G), and $\(\delta(G)>\frac{(b-1)p+a+b+bn-2}{a+b-1}\)$. Furthermore, it is shown that the result in this paper is best possible in some sense.
Details
- ISSN :
- 1469509X and 00170895
- Volume :
- 52
- Database :
- OpenAIRE
- Journal :
- Glasgow Mathematical Journal
- Accession number :
- edsair.doi...........c4cbd371219a831c064cc56b3fad4d5c