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A note on the three color problem. (English) Zbl 0824.05023
In 1990 Paul Erdős asked: Is there an integer \(k\geq 5\) such that if \(G\) is a planar graph without \(i\)-circuits, \(4\leq i\leq k\), then \(G\) is 3- colorable? A year later H. L. Abbott and B. Zhou answered this question affirmatively by proving the above with \(k= 11\). This note strengthens their result by showing that \(G\) is 3-colorable, if it is planar without \(i\)-circuits for \(i\) from 4 through 9.
Reviewer: M.Kubale (Gdańsk)

MSC:
05C15 Coloring of graphs and hypergraphs
05C10 Planar graphs; geometric and topological aspects of graph theory
05C38 Paths and cycles
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[2] Aksionov, V.A., Melnikov, L.S.: Essay on the theme: The three color problem. Combinatorics, Colloq. Math. Soc. János Bolyai18, 23–34 (1976)
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