Remarks on restricted fractional [formula omitted]-factors in graphs

Abstract

Assume there exists a function h:E(G)→[0,1] such that g(x)≤∑e∈E(G),x∋eh(e)≤f(x) for every vertex x of G. The spanning subgraph of G induced by the set of edges {e∈E(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∈M and h(e)=0 for each e∈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.