On the f -matching polytope and the fractional f -chromatic index
Stefan Glock · International Journal of Computer Mathematics · 2014
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 et al. [On the fractional f-chromatic index of a graph, Int. J. Comput. Math. 87 (2010), pp. 3359–3369] gave a new description of the f-matching polytope and thereby derived a formula for the fractional f-chromatic index stating that the fractional f-chromatic index is equal to 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.