Equitable colorings of bounded treewidth graphs
Theoretical Computer Science - Graph colorings
Journal of Combinatorial Theory Series B
Monadic Second Order Logic on Graphs with Local Cardinality Constraints
MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
Equitable colorings of Kronecker products of graphs
Discrete Applied Mathematics
Monadic second order logic on graphs with local cardinality constraints
ACM Transactions on Computational Logic (TOCL)
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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 Sm 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 $k \ge 16d$. 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.