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- W2369908970 abstract "Steinberg conjectured that every planar graph with neither cycle of length 4 nor cycle of length 5 is 3-colorable. Xu, independently, Borodin et al. proved that planar graphs without adjacent triangles and without 5- and 7-cycles are 3-colorable. As a corollary of this result, planar graphs without 4-, 5- and 7-cycles are 3-colorable. In this paper, we prove a result which is closer to the Steinbergs conjecture than the corollary mentioned above. Let G be a planar graph without 4- and 5-cycles. G is 3-colorable if it has no ( k , 7)-chords for each k ∈ {3, 6, 7}, where a ( k , 7)-chord is a chord of a cycle of length 7 + k -2 whose two ends divide the cycle into two paths, one of which is of length 6 and the other of length k -1." @default.
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- W2369908970 date "2013-04-01" @default.
- W2369908970 modified "2023-09-25" @default.
- W2369908970 title "A sufficient condition for a planar graph to be 3-colorable" @default.
- W2369908970 doi "https://doi.org/10.1360/012012-253" @default.
- W2369908970 hasPublicationYear "2013" @default.
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