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To the paper of H. L. Abbott and B. Zhou on 4-critical planar graphs. (English) Zbl 0881.05037
The following theorem is proved in this note: If a plane graph has minimum degree at least 3 and no triangles sharing an edge, then there is a face of size at least 4 and at most 10. This strengthens a result of H. L. Abbott and B. Zhou [Ars Comb. 32, 203-207 (1991; Zbl 0755.05062)], who proved that each 4-critical plane graph contains either a 4- or a 5-cycle or otherwise a face of size between 6 and 11.

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
05C15 Coloring of graphs and hypergraphs
05C38 Paths and cycles
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