DENSITIES, MATCHINGS, AND FRACTIONAL EDGE-COLORINGS

Given a multigraph G = (V, E) with a positive rational weight w(e) on each edge e, the weighted density problem (WDP) is to find a subset U of V, with vertical bar U vertical bar >= 3 and odd, that maximizes 2w(U)/(vertical bar U vertical bar - 1), where w(U) is the total weight of all edges with both ends in U, and the weighted fractional edge-coloring problem can be formulated as the following linear program: minimize 1(T) x subject to Ax = w, x >= 0, where A is the edge-matching incidence matrix of G. These two problems are closely related to the celebrated Goldberg-Seymour conjecture on edge-colorings of multigraphs, and are interesting in their own right. Even when w(e) = 1 for all edges e, determining whether WDP can be solved in polynomial time was posed by Jensen and Toft [Topics in Chromatic Graph Theory, Cambridge University Press, Cambridge, 2015, pp. 327-357] and by Stiebitz et al. [Graph Edge Colouring: Vizing's Theorem and Goldberg's Conjecture, John Wiley, New York, 2012] as an open problem. In this paper we present strongly polynomial-time algorithms for solving them exactly, and develop a novel matching removal technique for multigraph edge-coloring.