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 Δ 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 Δ acts in the results. Furthermore, counterexamples are structured to reveal the sharpness of degree conditions in the setting f ( x ) = g ( x ) + Δ.

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