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


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 \(\varDelta +1\).


Planar graph Total coloring Cycle Independent set 



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.


  1. Bojarshinov VA (2001) Edge and total coloring of interal graphs. Discret Appl Math 114:23–28MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bondy JA, Murty USR (1976) Graph theory with applications. MacMillan, LondonCrossRefzbMATHGoogle Scholar
  3. Du DZ, Shen L, Wang YQ (2009) Planar graphs with maximum degree 8 and without adjacent triangles are 9-totally-colorable. Discret Appl Math 157:2778–2784MathSciNetCrossRefzbMATHGoogle Scholar
  4. Kostochka AV (1996) The total chromatic number of any multigraph with maximum degree five is at most seven. Discret Math 162:199–214MathSciNetCrossRefzbMATHGoogle Scholar
  5. Kowalik L, Sereni J-S, S̆krekovski R (2008) Total-coloring of plane graphs with maximum degree nine. SIAM J Discret Math 22:1462–1479MathSciNetCrossRefzbMATHGoogle Scholar
  6. Liu B, Hou JF, Wu JL, Liu GZ (2009) Total colorings and list total colorings of planar graphs without intersecting 4-cycles. Discret Math 309:6035–6043MathSciNetCrossRefzbMATHGoogle Scholar
  7. McDiarmid CJH, Snchez-Arroyo A (1994) Total colorings regular bipartite graphs is NP-hard. Discret Math 124:155–162CrossRefzbMATHGoogle Scholar
  8. Roussel N, Zhu X (2010) Total coloring of planar graphs of maximum degree eight. Inf Process Lett 110:321–324MathSciNetCrossRefzbMATHGoogle Scholar
  9. Sanchez-Arroyo A (1989) Determing the total colouring number is NP-hard. Discret Math 78:315–319MathSciNetCrossRefzbMATHGoogle Scholar
  10. Sanders DP, Zhao Y (1999) On total 9-coloring planar graphs of maximum degree seven. J Graph Theory 31:67–73MathSciNetCrossRefzbMATHGoogle Scholar
  11. Shen L, Wang YQ (2009) Total colorings of planar graphs with maximum degree at least 8. Sci China Ser A 52:1733–1742MathSciNetCrossRefzbMATHGoogle Scholar
  12. Tan X, Chen HY, Wu JL (2009) Total colorings of planar graphs without adjacent 4-cycles. In: The eighth international symposium on operations research and its applications (ISORA’09), 167–173Google Scholar
  13. Wang GH, Yan GY, Yu JG, Zhang X (2013) The \(r\)-acyclic chromatic number of planar graphs. J Comb Optim. doi: 10.1007/s10878-013-9680-2
  14. Wang HJ, Wu LD, Wu WL (2014) Total coloring of planar graphs with maximum degree 8. Theor Comput Sci 522:54–61MathSciNetCrossRefzbMATHGoogle Scholar
  15. Wang HJ, Wu LD, Wu WL, Pardalos PM, Wu JL (2014) Minimum total coloring of planar graph. J Global Optim 60:777–791MathSciNetCrossRefzbMATHGoogle Scholar
  16. Wu JL (2004) Total coloring of series-parallel graphs. Ars Comb 73:215–217MathSciNetzbMATHGoogle Scholar
  17. Wu JL, Wang P (2008) List-edge and list-total colorings of graphs embedded on hyperbolic surfaces. Discret Math 308:6210–6215MathSciNetCrossRefzbMATHGoogle Scholar
  18. Yap HP (1996) Total colorings of graphs. Lecture notes in mathematics, vol 1623. Springer-Verlag, BerlinGoogle Scholar

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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