A Characterization of Graphs Having All (g, f)-Factors

LetGbe a graph with vertex setVand letg, f:V→Z+. We say thatGhas all (g, f)-factors ifGhas anh-factor for everyh:V→Z+such thatg(v)⩽h(v)⩽f(v) for everyv∈Vand at least one suchhexists. In this note, we derive from Tutteʹsf-factor theorem a similar characterization for the property of having all (g, f)-factors. An analogous result for parity-factors is presented also.