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Planar graphs without cycles of length from 4 to 7 are 3-colorable. (English) Zbl 1056.05052
Summary: Planar graphs without cycles of length from 4 to 7 are proved to be 3-colorable. Moreover, it is proved that each proper 3-coloring of a face of length from 8 to 11 in a connected plane graph without cycles of length from 4 to 7 can be extended to a proper 3-coloring of the whole graph. This improves on the previous results on a long standing conjecture of Steinberg.

MSC:
05C15 Coloring of graphs and hypergraphs
05C10 Planar graphs; geometric and topological aspects of graph theory
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