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Journal of Combinatorial Optimization

, Volume 33, Issue 1, pp 265–274 | Cite as

Total coloring of planar graphs without adjacent short cycles

  • Huijuan Wang
  • Bin LiuEmail author
  • Yan Gu
  • Xin Zhang
  • Weili Wu
  • Hongwei Gao
Article

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 \(G=(V,E)\) be a graph. Total coloring is a coloring of \(V\cup {E}\) such that no two adjacent or incident elements (vertex/edge) receive the same color. Let G be a planar graph with \(\varDelta \ge 8\). We proved that if for every vertex \(v\in V\), there exists two integers \(i_v,j_v\in \{3,4,5,6,7\}\) such that v is not incident with adjacent \(i_v\)-cycles and \(j_v\)-cycles, then the total chromatic number of graph G is \(\varDelta +1\).

Keywords

Planar graph Total coloring Cycle Independent set 

Notes

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grants 11301410, 11401386, 11402075, 11501316, 71171120, 71571108, the Projects of International (Regional) Cooperation and Exchanges of NSFC (71411130215), the Specialized Research Fund for the Doctoral Program of Higher Education (20133706110002), China Postdoctoral Science Foundation under Grants 2015M570568, 2015M570572, and the Shandong Provincial Natural Science Foundation of China under Grants ZR2014AQ001, ZR2015GZ007.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Huijuan Wang
    • 1
  • Bin Liu
    • 2
    Email author
  • Yan Gu
    • 1
  • Xin Zhang
    • 3
  • Weili Wu
    • 4
    • 5
  • Hongwei Gao
    • 1
  1. 1.College of MathematicsQingdao UniversityQingdaoChina
  2. 2.Department of MathematicsOcean University of ChinaQingdaoChina
  3. 3.School of Mathematics and StatisticsXidian UniversityXi’anChina
  4. 4.College of Computer Science and TechnologyTaiYuan University of TechnologyTaiyuanChina
  5. 5.Department of Computer ScienceUniversity of Texas at DallasRichardsonUSA

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