Abstract
Vizing and Behzad independently conjectured that every graph is (Δ + 2)-totally-colorable, where Δ denotes the maximum degree of G. This conjecture has not been settled yet even for planar graphs. The only open case is Δ = 6. It is known that planar graphs with Δ ≥ 9 are (Δ + 1)-totally-colorable. We conjecture that planar graphs with 4 ≤ Δ ≤ 8 are also (Δ + 1)-totally-colorable. In addition to some known results supporting this conjecture, we prove that planar graphs with Δ = 6 and without 4-cycles are 7-totally-colorable.
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Supported by the Natural Science Foundation of Department of Education of Zhejiang Province, China, Grant No. 20070441.
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Shen, L., Wang, Y. On the 7 Total Colorability of Planar Graphs with Maximum Degree 6 and without 4-cycles. Graphs and Combinatorics 25, 401–407 (2009). https://doi.org/10.1007/s00373-009-0843-y
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DOI: https://doi.org/10.1007/s00373-009-0843-y