A note concerning graphs with unique f-factors

We show that if a 2-edge connected graph G has a unique f-factor F, then some vertex has the same degree in F as in G. This conclusion is the best possible, even if the hypothesis is considerably strengthened.

[1]  Michael D. Plummer,et al.  On the 1-factors of a non-separable graph , 1967 .

[2]  W. Mader,et al.  1-Faktoren von Graphen , 1973 .

[3]  Roger C. Entringer,et al.  Spanning cycles of nearly cubic graphs , 1980, J. Comb. Theory B.

[4]  W. T. Tutte.  A Short Proof of the Factor Theorem for Finite Graphs , 1954, Canadian Journal of Mathematics.

[5]  G. R. T. Hendry.  Maximum graphs with a unique k-factor , 1984, J. Comb. Theory, Ser. B.