Nash-Williams conditions for the existence of all fractional [a, b]-factors

Zhiren Sun () and Sizhong Zhou ()
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Zhiren Sun: Nanjing Normal University
Sizhong Zhou: Jiangsu University of Science and Technology

Indian Journal of Pure and Applied Mathematics, 2021, vol. 52, issue 2, 542-547

Abstract: Abstract Let a and b be two positive integers with with $$1\le a\le b$$ 1 ≤ a ≤ b and $$b\ge 2$$ b ≥ 2 , and let G be a graph of order $$n\ge \frac{(a+b-3)(2a+b-1)-a+1}{a}$$ n ≥ ( a + b - 3 ) ( 2 a + b - 1 ) - a + 1 a . We say that G has all fractional [a, b]-factors if G has a fractional r-factor for any integer r with $$a\le r\le b$$ a ≤ r ≤ b . In this paper, we testify that $$G-I$$ G - I has all fractional [a, b]-factors for any independent set I of G if $$\delta (G)\ge \frac{(a+b-1)n+a+b-1}{2a+b-1}$$ δ ( G ) ≥ ( a + b - 1 ) n + a + b - 1 2 a + b - 1 and $$\delta (G)>\frac{(a+b-2)n+2\alpha (G)-1}{2a+b-2}$$ δ ( G ) > ( a + b - 2 ) n + 2 α ( G ) - 1 2 a + b - 2 . Furthermore, the result in Theorem 1 is sharp.

Keywords: Graph; Stability number; Minimum degree; Fractional [a; b]-factor; All fractional [a; b]-factors; 05C70 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s13226-021-00054-3

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