Total coloring of planar graphs without short cycles

Hua Cai, Jianliang Wu () and Lin Sun
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Hua Cai: Shandong University
Jianliang Wu: Shandong University
Lin Sun: Shandong University

Journal of Combinatorial Optimization, 2016, vol. 31, issue 4, No 19, 1650-1664

Abstract: Abstract The total chromatic number of a graph $$G$$ G , denoted by $$\chi ''(G)$$ χ ′ ′ ( G ) , is the minimum number of colors needed to color the vertices and edges of $$G$$ G such that no two adjacent or incident elements get the same color. It is known that if a planar graph $$G$$ G has maximum degree $$\Delta (G)\ge 9$$ Δ ( G ) ≥ 9 , then $$\chi ''(G)=\Delta (G)+1$$ χ ′ ′ ( G ) = Δ ( G ) + 1 . In this paper, it is proved that if $$G$$ G is a planar graph with $$\Delta (G)\ge 7$$ Δ ( G ) ≥ 7 , and for each vertex $$v$$ v , there is an integer $$k_v\in \{3,4,5,6,7,8\}$$ k v ∈ { 3 , 4 , 5 , 6 , 7 , 8 } such that there is no $$k_v$$ k v -cycle which contains $$v$$ v , then $$\chi ''(G)=\Delta (G)+1$$ χ ′ ′ ( G ) = Δ ( G ) + 1 .

Keywords: Planar graph; Total coloring; Cycle (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s10878-015-9859-9

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