146 research outputs found

    Minimum Sum Edge Colorings of Multicycles

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    In the minimum sum edge coloring problem, we aim to assign natural numbers to edges of a graph, so that adjacent edges receive different numbers, and the sum of the numbers assigned to the edges is minimum. The {\em chromatic edge strength} of a graph is the minimum number of colors required in a minimum sum edge coloring of this graph. We study the case of multicycles, defined as cycles with parallel edges, and give a closed-form expression for the chromatic edge strength of a multicycle, thereby extending a theorem due to Berge. It is shown that the minimum sum can be achieved with a number of colors equal to the chromatic index. We also propose simple algorithms for finding a minimum sum edge coloring of a multicycle. Finally, these results are generalized to a large family of minimum cost coloring problems

    Proof of the Goldberg-Seymour Conjecture on Edge-Colorings of Multigraphs

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    Given a multigraph G=(V,E)G=(V,E), the {\em edge-coloring problem} (ECP) is to color the edges of GG with the minimum number of colors so that no two adjacent edges have the same color. This problem can be naturally formulated as an integer program, and its linear programming relaxation is called the {\em fractional edge-coloring problem} (FECP). In the literature, the optimal value of ECP (resp. FECP) is called the {\em chromatic index} (resp. {\em fractional chromatic index}) of GG, denoted by χ(G)\chi'(G) (resp. χ(G)\chi^*(G)). Let Δ(G)\Delta(G) be the maximum degree of GG and let Γ(G)=max{2E(U)U1:UV,U3andodd},\Gamma(G)=\max \Big\{\frac{2|E(U)|}{|U|-1}:\,\, U \subseteq V, \,\, |U|\ge 3 \hskip 2mm {\rm and \hskip 2mm odd} \Big\}, where E(U)E(U) is the set of all edges of GG with both ends in UU. Clearly, max{Δ(G),Γ(G)}\max\{\Delta(G), \, \lceil \Gamma(G) \rceil \} is a lower bound for χ(G)\chi'(G). As shown by Seymour, χ(G)=max{Δ(G),Γ(G)}\chi^*(G)=\max\{\Delta(G), \, \Gamma(G)\}. In the 1970s Goldberg and Seymour independently conjectured that χ(G)max{Δ(G)+1,Γ(G)}\chi'(G) \le \max\{\Delta(G)+1, \, \lceil \Gamma(G) \rceil\}. Over the past four decades this conjecture, a cornerstone in modern edge-coloring, has been a subject of extensive research, and has stimulated a significant body of work. In this paper we present a proof of this conjecture. Our result implies that, first, there are only two possible values for χ(G)\chi'(G), so an analogue to Vizing's theorem on edge-colorings of simple graphs, a fundamental result in graph theory, holds for multigraphs; second, although it is NPNP-hard in general to determine χ(G)\chi'(G), we can approximate it within one of its true value, and find it exactly in polynomial time when Γ(G)>Δ(G)\Gamma(G)>\Delta(G); third, every multigraph GG satisfies χ(G)χ(G)1\chi'(G)-\chi^*(G) \le 1, so FECP has a fascinating integer rounding property

    Colorful Strips

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    Given a planar point set and an integer kk, we wish to color the points with kk colors so that any axis-aligned strip containing enough points contains all colors. The goal is to bound the necessary size of such a strip, as a function of kk. We show that if the strip size is at least 2k12k{-}1, such a coloring can always be found. We prove that the size of the strip is also bounded in any fixed number of dimensions. In contrast to the planar case, we show that deciding whether a 3D point set can be 2-colored so that any strip containing at least three points contains both colors is NP-complete. We also consider the problem of coloring a given set of axis-aligned strips, so that any sufficiently covered point in the plane is covered by kk colors. We show that in dd dimensions the required coverage is at most d(k1)+1d(k{-}1)+1. Lower bounds are given for the two problems. This complements recent impossibility results on decomposition of strip coverings with arbitrary orientations. Finally, we study a variant where strips are replaced by wedges
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