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Circular coloring of graphs via linear programming and tabu search

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Abstract

Circular coloring is a popular branch of graph theory which has been exhaustively studied for two decades mainly from a theoretical perspective. Since it is a refinement of the traditional proper coloring, it provides a more accurate model for cyclic scheduling problems which often arise in industrial applications. The present paper briefly surveys a special class of open shop scheduling that can be solved via circular coloring, and then proposes a new mathematical programming model and tabu search algorithm to compute the circular chromatic number of a graph effectively.

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Notes

  1. This part of the argument is the same as in Zhu (2001). The existence of the claimed finite \(G'\) follows by the ‘compactness principle’; see e.g. Truszczyński and Zs Tuza (1992) for its various formulations and many applications.

  2. Depending on the vertex order under consideration, the number of essential constraints may be smaller: if \(v_i^k\) and \(v_j^{k+1}\) are adjacent then the corresponding line of (2) automatically follows from that of (5); and if \(j,k,l,m\) are four distinct indices such that \(v_jv_k,v_lv_m\in E(G)\) and both of \(v_j,v_k\) precede both of \(v_l,v_m\) then only (5) is necessary for them. The same simplification applies in the situation where \(v_k\) and \(v_l\) coincide. As a consequence, each vertex has to be involved in at most two constraints of type (5)–(6).

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Acknowledgments

Research of the second author was supported in part by the Hungarian Scientific Research Fund, OTKA grant 81493.

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Authors and Affiliations

  1. Department of Computer Science and Systems Technology, University of Pannonia, Egyetem u. 10, Veszprém, 8200, Hungary

    Mate Barany & Zsolt Tuza

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13–15, Budapest, 1053, Hungary

    Zsolt Tuza

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  1. Mate Barany
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  2. Zsolt Tuza
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Correspondence to Mate Barany.

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Barany, M., Tuza, Z. Circular coloring of graphs via linear programming and tabu search. Cent Eur J Oper Res 23, 833–848 (2015). https://doi.org/10.1007/s10100-014-0345-8

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