Geodesic
On the complexity for constructing a 3-colouring for planar graphs with short facets
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 20 (2018) no. 2, pp. 199-205

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The vertex 3-colourability problem is to determine for a given graph whether one can divide its vertex set into three subsets of pairwise non-adjacent vertices. This problem is NP-complete in the class of planar graphs, but it becomes polynomial-time solvable for planar triangulations, i.e. planar graphs, all facets of which (including external) are triangles. Additionally, the problem is NP-complete for planar graphs whose vertices have degrees at most 4, but it becomes linear-time solvable for graphs whose vertices have maximal degree at most 3. So it is an interesting question to find a threshold for lengths of facets and maximum vertex degree, for which the complexity of the vertex 3-colourability changes from polynomial-time solvability to NP-completeness. In this paper we answer this question and prove NP-completeness of the vertex 3-colourability problem in the class of planar graphs of the maximum vertex degree at most 5, whose facets are triangles and quadrangles only.
Keywords: computational complexity, vertex 3-colourability problem, planar graph. 
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     author = {D. V. Sirotkin},
     title = {On the complexity for constructing a 3-colouring for planar graphs with short facets},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {199--205},
     publisher = {mathdoc},
     volume = {20},
     number = {2},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2018_20_2_a5/}
}
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D. V. Sirotkin. On the complexity for constructing a 3-colouring for planar graphs with short facets. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 20 (2018) no. 2, pp. 199-205. http://geodesic.mathdoc.fr/item/SVMO_2018_20_2_a5/