International audienceIn this article we consider the problem of gathering information in a gateway in a radio mesh access network. Due to interferences, calls (transmissions) cannot be performed simultaneously. This leads us to define a round as a set of non-interfering calls. Following the work of Klasing, Morales and Pérennes, we model the problem as a Round Weighting Problem (RWP) in which the objective is to minimize the overall period of non-interfering calls activations (total number of rounds) providing enough capacity to satisfy the throughput demand of the nodes.We develop tools to obtain lower and upper bounds for general graphs. Then, more precise results are obtained considering a symmetric interference model based on distance of graphs, called the distance-d interference model (the particular case d= 1 corresponds to the primary node model).We apply the presented tools to get lower bounds for grids with thegateway either in the middle or in the corner. We obtain upper boundswhich in most of the cases match the lower bounds, using strategiesthat either route the demand of a single node or route simultaneously flow from several source nodes. Therefore, weobtain exact and constructive results for grids, in particularfor the case of uniform demands answering a problem asked by Klasing, Morales and Pérennes

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a b s t r a c tMotivated by bandwidth allocation under interference constraints in radio networks, we define and investigate an optimization problem that combines the classical flow and edge coloring problems in graphs. Let G = (V , E) be a graph with a demand function b : V → Z + and a gateway gis a set with one flow for each source node. Every flow φ defines a multigraph G φ with vertex set V and all edges in the paths in φ. A distance-d edge coloring of a flow φ is an edge coloring of G φ such that two edges with the same color are at distance at least d in G. The distance-d flow coloring problem (FCP d ) is the problem of obtaining a flow φ on G with a distance-d edge coloring where the number of used colors is minimum. For any fixed d ≥ 3, we prove that FCP d is NP-hard even on a bipartite graph with just one source node. For d = 2, we also prove NP-hardness on a bipartite graph with multiples sources. For d = 1, we show that the problem is polynomial in 3-connected graphs and bipartite graphs. Finally, we show that a list version of the problem is inapproximable in polynomial time by a factor of O(log n) even on n-vertex paths, for any d ≥ 1.

A blink is a plane graph with a bipartition (black, gray) of its edges. Subtle classes of blinks are in 1-1 correspondence with closed, oriented and connected 3-manifolds up to orientation preserving homeomorphisms [14]. Switching black and gray in a blink B, giving −B, reverses the manifold orientation. The dual of the blink B in the sphere S 2 is denoted by B . Blinks B and −B induce the same 3-manifold. The paper reinforces the Conjecture that if B / ∈ {B, −B }, then the monochromatic 3-connected (mono3c) blinks B and B induce distinct 3-manifolds. Using homology of covers and length spectra, we conclude the topological classification of 708 mono3c blinks that were organized in equivalence classes by . We also present a reformulation of the combinatorial algorithm to obtain the WRT-invariants of [13] using only the blink.