Total coloring of planar graphs without adjacent short cycles

Huijuan Wang (), Bin Liu (), Yan Gu (), Xin Zhang (), Weili Wu () and Hongwei Gao ()
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Huijuan Wang: Qingdao University
Bin Liu: Ocean University of China
Yan Gu: Qingdao University
Xin Zhang: Xidian University
Weili Wu: TaiYuan University of Technology
Hongwei Gao: Qingdao University

Journal of Combinatorial Optimization, 2017, vol. 33, issue 1, No 18, 265-274

Abstract: Abstract In the study of computer science, optimization, computation of Hessians matrix, graph coloring is an important tool. In this paper, we consider a classical coloring, total coloring. Let $$G=(V,E)$$ G = ( V , E ) be a graph. Total coloring is a coloring of $$V\cup {E}$$ V ∪ E such that no two adjacent or incident elements (vertex/edge) receive the same color. Let G be a planar graph with $$\varDelta \ge 8$$ Δ ≥ 8 . We proved that if for every vertex $$v\in V$$ v ∈ V , there exists two integers $$i_v,j_v\in \{3,4,5,6,7\}$$ i v , j v ∈ { 3 , 4 , 5 , 6 , 7 } such that v is not incident with adjacent $$i_v$$ i v -cycles and $$j_v$$ j v -cycles, then the total chromatic number of graph G is $$\varDelta +1$$ Δ + 1 .

Keywords: Planar graph; Total coloring; Cycle; Independent set (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10878-015-9954-y

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