A sufficient condition for planar graphs with maximum degree 8 to be 9-totally colorable |
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| Authors: | Jian Sheng Cai Chang Chun Teng Gui Ying Yan |
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| Affiliation: | 1. School of Mathematics and Information Sciences, Weifang University, Weifang, 261061, P. R. China
2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, P. R. China
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| Abstract: | A total k-coloring of a graph G is a coloring of V (G) ∪ E(G) using k colors such that no two adjacent or incident elements receive the same color. The total chromatic number χ″(G) is the smallest integer k such that G has a total k-coloring. It is known that if a planar graph G has maximum degree Δ ≥ 9, then χ″(G) = Δ+1. In this paper, we prove that if G is a planar graph with maximum degree 8 and without a fan of four adjacent 3-cycles, then χ″(G) = 9. |
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| Keywords: | Total coloring planar graph a fan of four adjacent -cycles |
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