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An equitable graph coloring is a proper vertex coloring of a graph G wherethe sizes of the color classes differ by at most one. The equitable chromaticnumber is the smallest number k such that G admits such equitable k-coloring.We focus on enumerative algorithms for the computation of the equitablecoloring number and propose a general scheme to derive pruning rules for them:We show how the extendability of a partial coloring into an equitable coloringcan be modeled via network flows. Thus, we obtain pruning rules which can bechecked via flow algorithms. Computational experiments show that the searchtree of enumerative algorithms can be significantly reduced in size by theserules and, in most instances, such naive approach even yields a fasteralgorithm. Moreover, the stability, i.e., the number of solved instances withina given time limit, is greatly improved.  Since the execution of flow algorithms at each node of a search tree is timeconsuming, we derive arithmetic pruning rules (generalized Hall-conditions)from the network model. Adding these rules to an enumerative algorithm yieldsan even larger runtime improvement.

A flow based pruning scheme for enumerative equitable coloring algorithms