Existence of all generalized fractional [formula omitted]-factors of graphs

Abstract

Let G be a graph and R+ denote the set of non-negative real numbers. For a vertex v of G, EG(v) denotes the set of edges incident with v. Let φ:E(G)→R+ and f:V(G)→R+. Then a generalized fractional f-factor of G is a real-valued function ω:E(G)→R+ that satisfies 0≤ω(e)≤φ(e) for every e∈E(G) and f(v)=∑e∈EG(v)ω(e) for every v∈V(G). For two functions g,f:V(G)→R+ with g≤f, we say that G has all generalized fractional (g,f)-factors if G has a generalized fractional h-factor for every h:V(G)→R+ satisfying g(x)≤h(x)≤f(x) for all x∈V(G). In this paper, we present a necessary and sufficient condition for a graph G to have all generalized fractional (g,f)-factors, and moreover, our proof is self-contained and does not use the (g,f)-factor theorem or the fractional (g,f)-factor theorem.