On the f-matching polytope and the fractional f-chromatic index

Our motivation is the question how similar the f -colouring problem is to the classic edge-colouring problem, particularly with regard to graph parameters. In 2010, Zhang, Yu, and Liu [9] gave a new description of the f -matching polytope and derived a formula for the fractional f -chromatic index, stating that the fractional f -chromatic index equals the maximum of the fractional maximum f -degree and the fractional f -density. Unfortunately, this formula is incorrect. We present counterexamples for both the description of the f -matching polytope and the formula for the fractional f -chromatic index. Finally, we prove a short lemma concerning the generalization of Goldberg's Conjecture.