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On equitable coloring of d-degenerate graphs
Journal article   Peer reviewed

On equitable coloring of d-degenerate graphs

A V Kostochka, K Nakprasit and S V Pemmaraju
SIAM journal on discrete mathematics, Vol.19(1), pp.83-95
01/01/2005
DOI: 10.1137/S0895480103436505

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Abstract

An equitable coloring of a graph is a proper vertex coloring such that the sizes of any two color classes differ by at most 1. A d-degenerate graph is a graph G in which every subgraph has a vertex with degree at most d. A star S-m with m rays is an example of a 1-degenerate graph with maximum degree m that needs at least 1 + m/2 colors for an equitable coloring. Our main result is that every n-vertex d-degenerate graph G with maximum degree at most n/15 can be equitably k-colored for each k >= 16d. The proof of this bound is constructive. We extend the algorithm implied in the proof to an O(d)-factor approximation algorithm for equitable coloring of an arbitrary d-degenerate graph. Among the implications of this result is an O( 1)-factor approximation algorithm for equitable coloring of planar graphs with fewest colors. A variation of equitable coloring ( equitable partitions) is also discussed.
Mathematics Mathematics, Applied Physical Sciences Science & Technology

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