We present new min-max relations in digraphs between the number of paths satisfying certain conditions and the order of the corresponding cuts. We define these objects in order to capture, in the context of solving the half-integral linkage problem, the essential properties needed for reaching a large bramble of constant congestion from the terminal set. This strategy has been used ad-hoc in several articles, usually with lengthy technical proofs, and our objective is to abstract it to make it applicable in a simpler and unified way. We provide two proofs of the min-max relations, one consisting in applying Menger’s Theorem on appropriately defined digraphs, and an alternative simpler one using matroids, however with worse polynomial running time. As an application, we manage to simplify and improve several results of Edwards et al. in 2017 and of Giannopoulou et al. in 2022 about finding half-integral linkages in digraphs. Concerning the former, besides being simpler, our proof provides an almost optimal bound on the strong connectivity of a digraph for it to be half-integrally feasible under the presence of a large bramble of congestion two (or equivalently, if the directed tree-width is large). Concerning the latter, our proof uses brambles as rerouting objects instead of cylindrical grids, hence yielding much better bounds and being somehow independent of a particular topology. We hope that our min-max relations will find further applications as, in our opinion, they are simple, robust, and versatile to be easily applicable to different types of routing problems in digraphs.

A b-greedy colouring is a colouring which is both a b-colouring and a greedy colouring. A z-colouring is a b-greedy colouring such that a b-vertex of the largest colour is adjacent to a b-vertex of every other colour. The b-Grundy number (resp. z-number) of a graph is the maximum number of colours in a b-greedy colouring (resp. z-colouring) of it. In this paper, we study those two parameters. We show that similarly to the z-number, the b-Grundy number is not monotone and can be arbitrarily smaller than the minimum of the Grundy number and the b-chromatic number. We also describe a polynomial-time algorithm that decides whether a given k-regular graph has b-Grundy number (resp. z-number) equal to k+1. We also prove that every cubic graph with no induced 4-cycle has b-Grundy number and z-number exactly 4.

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