# A flow based pruning scheme for enumerative equitable coloring algorithms

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## Abstract

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

## Notes

### Acknowledgements

This work is partially supported by the German Federal Ministry of Education and Research (BMBF Grant No. 05M13PAA, joint project 05M2013-VINO: Virtual Network Optimization). Moreover, it was supported by an RWTH Aachen Seed Fund project and the RWTH Aachen Undergraduate Funds, both funded by the Excellence Initiative of the German Federal and State Governments.

We thank our student assistants Sven Förster and Duc Thanh Tran for their work, especially regarding implementations and testing.

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