Local condition for planar graphs of maximum degree 7 to be 8-totally colorable |
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| Authors: | Gerard Jennhwa Chang Jianfeng Hou |
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| Institution: | a National Taiwan University, Department of Mathematics, Taipei 10617, Taiwanb Institute for Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwanc National Center for Theoretical Sciences, Taipei, Taiwand Center for Discrete Mathematics, Fuzhou University, Fuzhou, Fujian, 350002, China |
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| Abstract: | The total chromatic number of a graph G, denoted by χ″(G), is the minimum number of colors needed to color the vertices and edges of G such that no two adjacent or incident elements get the same color. 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 7, and for every vertex v, there is an integer kv∈{3,4,5,6} so that v is not incident with any kv-cycle, then χ″(G)=8. |
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| Keywords: | Total coloring Total chromatic number Planar graphs Cycle |
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