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- W2029120932 abstract "Let G be a simple undirected graph and let w be an assignment of non-negative weights to the vertices of G. For a proper r-coloring c of the vertices of G, we denote by wc(i) the maximum weight of a vertex in color class i, and denote by wc the sum, ∑i = 1r wc(i), of these maximum weights. Let σ(G, w; r) be the minimum of wc among all r-colorings of G, and let σ(G, w) be the minimum among σ(G, w; r) for all r ⩾ 1. A coloring c of a weighted graph G is called optimal coloring if wc = σ(G, w). We shall consider the problem that how many colors are needed in an optimal coloring of a weighted graph G. We relate this number to the First-Fit coloring algorithm and show that for optimal colorings of trees, the number of colors needed could be arbitrarily large, and for graphs of path-width k, the number of colors needed in an optimal coloring is at most 40(k + 1). We then give a polynomial time algorithm to decide, for a fixed r, whether or not a weighted graph G of bounded tree-width satisfies σ(G, w; r) ⩽ q, for any given number q." @default.
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- W2029120932 date "1997-01-01" @default.
- W2029120932 modified "2023-10-17" @default.
- W2029120932 title "A coloring problem for weighted graphs" @default.
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- W2029120932 doi "https://doi.org/10.1016/s0020-0190(97)00002-1" @default.
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