The Flow Coloring Problem∗

Suppose a graph G = (V,E) with one destination node g (the gateway) and a set of source nodes with integer demands de ning the input of the problem. Let Φ stand for the set of all possible ows φ : E → Z+ sending all demands from the sources to the gateway. Then each φ ∈ Φ de nes a multigraph Gφ = (V,E, φ), where φ represents the multiplicity of the edges. In the ow coloring problem, the objective is to nd the ow chromatic index χΦ(G) = minφ∈Φ χ (Gφ), where χ (Gφ) is the chromatic index of Gφ. We relate χ ′ Φ(G) to the ow fractional chromatic index χΦ,f (G) = minφ∈Φ χ ′ f (Gφ), where χ ′ f (Gφ) is the fractional chromatic index of Gφ. We are interested in proving that the inequality χ ′ Φ,f (G) ≤ χΦ(G) ≤ χΦ,f (G) + 1 is valid, following the classical Goldberg's Conjecture for arbitrary multigraphs. When G is 2-connected, we propose a polynomial algorithm to show that the inequality holds for several cases. Moreover, we prove that this algorithm is optimal for 3-connected graphs and gives a 3 2 -approximation for arbitrary 2-connected graphs. 1 Problem introduction Let G = (V,E) be a graph with a special node g ∈ V , to be called destination node or gateway. Each other node v ∈ V \g is associated with an integer demand bv ≥ 0 to be sent to g. We will call source node a node v with bv > 0. Let Φ stand for the set of all possible integer ows φ : E → Z+ sending the total demand from the sources to the gateway. Each φ ∈ Φ de nes a multigraph Gφ = (V,E, φ), where φ represents the multiplicity of the edges. In other words, the edge multiset of Gφ is de ned by each element e ∈ E replicated φ(e) times. The ow coloring problem (FCP) in G consists in nding the ow chromatic index χΦ(G, b) = minφ∈Φ χ (Gφ), where χ (Gφ) is the chromatic index of Gφ, i.e. the minimum number of colors assigned to the edges of Gφ such that every edge receives at least one color and no two edges with the same color meet at a node. Notice that an edge e ∈ E with multiplicity φ(e) = 0 does not appear in Gφ, so it does not need to be colored. When the vector of demands b has not a particular de nition, we simplify the notation by using χΦ(G) to denote χ ′ Φ(G, b). The edges of Gφ receiving the same color induces a matching in G. The number c(e) of matchings covering the edge e ∈ E is at least the ow φ(e) . Thus, c(e) can be seen as the capacity assigned to e. This observation leads to a restatement of the FCP as a minimum weighting of the matchings of G such that the sum of the (integer) weights of the matchings covering an edge de nes its capacity, and these capacities allow a ow sending the total demand from the sources to the gateway. We will say that the weighted matchings cover the ow. Actually, the term ow coloring can be used in more general contexts involving other combinations of ows (e.g. single or multi-commodity, single or multiple sources and destinations etc) and colorings (edge or node coloring, distance-d coloring meaning that nodes/edges at distance at most d cannot share a color). Moreover, either the ow or the coloring need not to be integer. Each possible combination leads to a variant of FCP. Particularly, we will relate χΦ(G, b) to the ow fractional chromatic index χΦ,f (G, b) = minφ∈Φ χ ′ f (Gφ), where χ ′ f (Gφ) is the fractional chromatic index of Gφ. Some scenarios of ow coloring have been studied in the literature under the name of Round Weighting Problem RWP [KMP08]. The coloring usually used in RWP is a kind of fractional edge ∗This work is partially supported by CNPq Universal and FUNCAP/CNPq PRONEM Projects. †Partially supported by CNPq, Brazil ‡C. Huiban is funded by FUNCAP/CNPq, Brazil.