Geodesic
A Neighborhood Condition for Fractional ID-[A, B]-Factor-Critical Graphs
Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 2, pp. 409-418

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Let G be a graph of order n, and let a and b be two integers with 1 ≤ a ≤ b. Let h : E(G) → [0, 1] be a function. If a ≤Σ_ e ∋ x h(e) ≤ b holds for any x ∈ V (G), then we call G[F_h] a fractional [a, b]-factor of G with indicator function h, where F_h = { e ∈ E(G) : h(e) gt; 0 }. A graph G is fractional independent-set-deletable [a, b]-factor-critical (in short, fractional ID-[a, b]-factor-critical) if G − I has a fractional [a, b]-factor for every independent set I of G. In this paper, it is proved that if n ≥(a+2b)(2a+2b-3)+1/b, δ (G) ≥bn/a+2b + a and | N_G(x) ∪ N_G(y) | ≥(a+b)n/a+2b for any two nonadjacent vertices x, y ∈ V (G), then G is fractional ID-[a, b]-factor-critical. Furthermore, it is shown that this result is best possible in some sense.
Keywords: graph, minimum degree, neighborhood, fractional [a, b]-factor, fractional ID-[a, b]-factor-critical graph 
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     title = {A {Neighborhood} {Condition} for {Fractional} {ID-[A,} {B]-Factor-Critical} {Graphs}},
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Zhou, Sizhong; Yang, Fan; Sun, Zhiren. A Neighborhood Condition for Fractional ID-[A, B]-Factor-Critical Graphs. Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 2, pp. 409-418. http://geodesic.mathdoc.fr/item/DMGT_2016_36_2_a11/