Remarks on restricted fractional (g, f )-factors in graphs

Assume there exists a function h : E(G) -> [0, 1] such that g(x) <= Sigma(e is an element of E(G),x is not an element of e) h(e) <= f(x) for every vertex x of G. The spanning subgraph of G induced by the set of edges {e E is an element of(G) : h(e) > 0} is called a fractional (g, f )-factor of G with indicator function h. Let M and N be two disjoint sets of independent edges of G satisfying |M| = m and |N| = n. We say that G possesses a fractional (g, f )-factor with the property E(m, n) if G contains a fractional (g, f )-factor with indicator function h such that h(e) = 1 for each e is an element of M and h(e) = 0 for each e is an element of N. In this article, we discuss stability number and minimum degree conditions for graphs to possess fractional (g, f )-factors with the property E(1, n). Furthermore, we explain that the stability number and minimum degree conditions declared in the main result are sharp. (c) 2022 Elsevier B.V. All rights reserved.