Abstract
The max-coloring problem is to compute a legal coloring of the vertices of a graph G=(V,E) with vertex weights w such that \(\sum_{i=1}^{k}\max_{v\in C_{i}}w(v_{i})\) is minimized, where C 1,…,C k are the various color classes. For general graphs, max-coloring is as hard as the classical vertex coloring problem, a special case of the former where vertices have unit weight. In fact, in some cases it can even be harder: for example, no polynomial time algorithm is known for max-coloring trees. In this paper we consider the problem of max-coloring paths and its generalization, max-coloring skinny trees, a broad class of trees that includes paths and spiders. For these graphs, we show that max-coloring can be solved in time O(|V|+time for sorting the vertex weights). When vertex weights are real numbers, we show a matching lower bound of Ω(|V|log |V|) in the algebraic computation tree model.
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T. Kavitha’s work done as part of the DST-MPG partner group “Efficient Graph Algorithms” at IISc Bangalore.
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Kavitha, T., Mestre, J. Max-coloring paths: tight bounds and extensions. J Comb Optim 24, 1–14 (2012). https://doi.org/10.1007/s10878-010-9290-1
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DOI: https://doi.org/10.1007/s10878-010-9290-1