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Planar graphs without 4, 6, 8-cycles are 3-colorable. (English) Zbl 1144.05033
It was conjectured by Steinberg in 1976 that every planar graph without 4-cycles and 5-cycles is 3-colorable [see T. Jensen and B. Toft, Graph Coloring Problems, New York, NY: John Wiley & Sons (1995; Zbl 0855.05054)] and it was asked by Erdős whether there exists an integer $$k$$ such that the absence of cycles with length from 4 to $$k$$ in a planar graph guarantees its 3-colorability. In this paper the authors prove that every planar graph without cycles of length 4, 6, and 8 is 3-colorable.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
##### Keywords:
planar graphs; coloring; cycle
Full Text:
##### References:
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