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

Our motivation is the question of 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 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.