Title: Linear programming formulation of the vertex colouring problem

Authors: Moustapha Diaby

Addresses: Operations and Information Management, University of Connecticut, Storrs, CT 06268, USA

Abstract: In this paper, we present a first linear programming (LP) formulation of the vertex colouring problem (VCP). The complexity orders of the number of variables and the number of constraints of the proposed LP are O (δ⁹·ς³) and O (δ⁸·ς³), respectively, where δ and ς are the number of vertices and the number of available colours in the VCP instance, respectively. Hence, the proposed model represents a reaffirmation of |P = NP|. First, we develop a bipartite network flow (BNF) based model of the problem. Then, we use a graph-based modelling framework similar to that of Diaby to develop the proposed LP model. A numerical example is used to illustrate the approach.

Keywords: vertex colouring; graph colouring; vertex packing; linear programming; combinatorial optimisation; computational complexity; bipartite network flow; graph-based modelling.

DOI: 10.1504/IJMOR.2010.032718

International Journal of Mathematics in Operational Research, 2010 Vol.2 No.3, pp.259 - 289

Published online: 17 Apr 2010 *

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