Abstract
Suppose that G is a planar graph with maximum degree Δ. In this paper it is proved that G is total-(Δ + 2)-choosable if (1) Δ ≥ 7 and G has no adjacent triangles (i.e., no two triangles are incident with a common edge); or (2) Δ ≥ 6 and G has no intersecting triangles (i.e., no two triangles are incident with a common vertex); or (3) Δ ≥ 5, G has no adjacent triangles and G has no k-cycles for some integer k ∈ {5, 6}.
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Supported by NSFC (Grant Nos. 10871119, 10971121), RFDP (Grant No. 200804220001) and NSFSP (Grant No. ZR2009AM009) of China
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Liu, B., Hou, J.F. & Liu, G.Z. List total colorings of planar graphs without triangles at small distance. Acta. Math. Sin.-English Ser. 27, 2437–2444 (2011). https://doi.org/10.1007/s10114-011-9154-3
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DOI: https://doi.org/10.1007/s10114-011-9154-3