Abstract.
This paper completes the constructive proof of the following result: Suppose p/q≥2 is a rational number, A is a finite set and f1,f2,···,f n are mappings from A to {0,1,···,p−1}. Then for any integer g, there is a graph G=(V,E) of girth at least g with such that G has exactly n (p,q)-colourings (up to equivalence) g1,g2,···,g n , and each g i is an extension of f i . A probabilistic proof of this result was given in [8]. A constructive proof of the case p/q≥3 was given in [7].
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This research was partially supported by the National Science Council under grant NSC91-2115-M-110-004
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Pan, Z., Zhu, X. Graphs of Large Girth with Prescribed Partial Circular Colourings. Graphs and Combinatorics 21, 119–129 (2005). https://doi.org/10.1007/s00373-004-0596-6
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DOI: https://doi.org/10.1007/s00373-004-0596-6