The Extension Degree Conditions for Fractional Factor

In Gao's previous work, the authors determined several degree conditions of a graph which admits fractional factor in particular settings. It was revealed that these degree conditions are tight if b = f(x) = g(x) = a for all vertices x in G. In this paper, we continue to discuss these degree conditions for admitting fractional factor in the setting that several vertices and edges are removed and there is a difference Delta between g(x) and f(x) for every vertex x in G. These obtained new degree conditions reformulate Gao's previous conclusions, and show how Delta acts in the results. Furthermore, counterexamples are structured to reveal the sharpness of degree conditions in the setting f(x) = g(x) + Delta.