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Determining the total colouring number is NP-hard. (English) Zbl 0695.05023
D. Leven and Z. Galil proved that [“NP-completeness of finding the chromatic index of regular graphs”, J. Algorithms 4, 35-44 (1983; Zbl 0509.68037)] the problem of determining the chromatic index of an arbitrary r-regular graph for $$r\geq 4$$ is NP-complete. The present author proves that a 4-regular graph G is 4-edge colourable if and only if another cubic bipartite graph H is 4-total colourable. Hence the problem of determining the total chromatic number of an arbitrary cubic bipartite graph is also NP-complete.
Reviewer: H.-P.Yap

##### MSC:
 05C15 Coloring of graphs and hypergraphs 68R10 Graph theory (including graph drawing) in computer science 68Q25 Analysis of algorithms and problem complexity
##### Keywords:
total colouring; edge-colouring; total chromatic number
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##### References:
 [1] Garey, M.; Johnson, D., Computers and intractability A guide to a NP-completeness theory, (1979), Freeman San Francisco · Zbl 0411.68039 [2] Holyer, I.J., The NP-completeness of edge colourings, SIAM J. computing, 10, 718-720, (1981) · Zbl 0473.68034 [3] Leven, D.; Galil, Z., NP-completeness of finding the chromatic index of regular graphs, J. algorithms, 4, 35-44, (1983) · Zbl 0509.68037
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