Skip to main content
SpringerLink
Log in

Total coloring of planar graphs without adjacent chordal 6-cycles

  • Published:
Journal of Combinatorial Optimization Aims and scope Submit manuscript

Abstract

A total coloring of a graph G is a coloring such that no two adjacent or incident elements receive the same color. In this field there is a famous conjecture, named Total Coloring Conjecture, saying that the the total chromatic number of each graph G is at most \(\Delta +2\). Let G be a planar graph with maximum degree \(\Delta \ge 7\) and without adjacent chordal 6-cycles, that is, two cycles of length 6 with chord do not share common edges. In this paper, it is proved that the total chromatic number of G is \(\Delta +1\), which partly confirmed Total Coloring Conjecture.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  • Bondy JA, Murty USR (1976) Graph theory with applications. MacMillan, London

    Book  MATH  Google Scholar 

  • Behzad M (1965) Graphs and their chromatic numbers. Ph.D. Thesis, Michigan State University

  • Borodin OV (1989) On the total coloring of planar graphs. J Reine Angew Math 394:180–185

    MathSciNet  MATH  Google Scholar 

  • Borodin OV, Kostochka AV, Woodall DR (1997) Total colorings of planar graphs with large maximum degree. J Graph Theory 26:53–59

    Article  MathSciNet  MATH  Google Scholar 

  • Cai H, Wu JL, Sun L (2016) Total coloring of planar graphs without short cycles. J Comb Optim 31:1650–1664

    Article  MathSciNet  MATH  Google Scholar 

  • Chang GJ, Hou JF, Roussel N (2011) Local condition for planar graphs of maximum degree 7 to be 8-totally colorable. Discret Appl Math 159:760–768

    Article  MathSciNet  MATH  Google Scholar 

  • Du DZ, Shen L, Wang YQ (2009) Planar graphs with maximum degree 8 and without adjacent triangles are \(9\)-total-colorable. Discret Appl Math 157:6035–6043

    Article  MATH  Google Scholar 

  • Hou JF, Liu B, Liu GZ, Wu JL (2011) Total colorings of planar graphs without 6-cycles. Discret Appl Math 159:157–163

    Article  MathSciNet  MATH  Google Scholar 

  • Kostochka AV (1996) The total chromatic number of any multigraph with maximum degree five is at most seven. Discret Math 162:199–214

    Article  MathSciNet  MATH  Google Scholar 

  • Kowalik L, Sereni JS, S̆krekovski R (2008) Total-coloring of plane graphs with maximum degree nine. SIAM J Discret Math 22:1462–1479

    Article  MathSciNet  MATH  Google Scholar 

  • Li H, Ding L, Liu B, Wang G (2015) Neighbor sum distinguishing total colorings of planar graphs. J Comb Optim 30:675–688

    Article  MathSciNet  MATH  Google Scholar 

  • Liu B, Hou JF, Wu JL, Liu GZ (2009) Total colorings and list total colorings of planar graphs without intersecting 4-cycles. Discret Math 309:6035–6043

    Article  MathSciNet  MATH  Google Scholar 

  • Qu C, Wang G, Wu J, Yu X (2016) On the neighbor sum distinguishing total coloring of planar graphs. Theor Comput Sci 609:162–170

    Article  MathSciNet  MATH  Google Scholar 

  • Qu C, Wang G, Yan G, Yu X (2015) Neighbor sum distinguishing total choosability of planar graphs. J Comb Optim. doi:10.1007/s10878-015-9911-9

  • Sanders DP, Zhao Y (1999) On total 9-coloring planar graphs of maximum degree seven. J Graph Theory 31:67–73

    Article  MathSciNet  MATH  Google Scholar 

  • Shen L, Wang YQ (2009) Total colorings of planar graphs with maximum degree at least 8. Sci China Ser A 52(8):1733–1742

    Article  MathSciNet  MATH  Google Scholar 

  • Vizing VG (1968) Some unsolved problems in graph theory. Uspekhi Mat Nauk 23:117–134 (in Russian)

    MathSciNet  MATH  Google Scholar 

  • Wang B, Wu JL (2011) Total coloring of planar graphs with maximum degree seven and without intersecting 3-cycles. Discret Math 311:2025–2030

    Article  MathSciNet  MATH  Google Scholar 

  • Wang B, Wu JL, Wang HJ (2014) Total coloring of planar graphs without chordal \(6\)-cycles. Discret Appl Math 171:116–121

    Article  MathSciNet  MATH  Google Scholar 

  • Wang HJ, Liu B, Gu Y, Zhang X, Wu WL, Gao HW (2015) Total coloring of planar graphs without adjacent short cycles, J Comb Optim. doi:10.1007/s10878-015-9954-y

  • Wang HJ, Luo ZY, Liu B, Gu Y, Gao HW (2016) A note on the minimum total coloring of planar graphs. Acta Math Sin Engl Ser 32:967–974

    Article  MathSciNet  MATH  Google Scholar 

  • Wang HJ, Wu LD, Wu WL, Pardalos PM, Wu JL (2014) Minimum total coloring of planar graph. J Glob Optim 60:777–791

    Article  MathSciNet  MATH  Google Scholar 

  • Wang P, Wu JL (2004) A note on the total colorings of planar graphs without \(4\)-cycles. Discret Math 24:125–135

    MathSciNet  MATH  Google Scholar 

  • Wang W (2007) Total chromatic number of planar graphs with maximum degree ten. J Graph Theory 54:91–102

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported in part by National Natural Science Foundation of China (11501316, 71171120, 71571108), State Scholarship Fund of China (201506335016), China Postdoctoral Science Foundation (2015M570568, 2016T90607), Shandong Provincial Natural Science Foundation of China (ZR2014AQ001, ZR2015GZ007, ZR2015FM023) and Qingdao Postdoctoral Application Research Project (2015170).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Qingdao University, Qingdao, 266071, China

    Huijuan Wang & Hongwei Gao

  2. School of Mathematical Sciences, Ocean University of China, Qingdao, 266100, China

    Bin Liu & Xiaoli Wang

  3. Department of Computer Science, University of Texas at Dallas, Richardson, TX, 75080, USA

    Guangmo Tong & Weili Wu

Authors
  1. Huijuan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bin Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaoli Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Guangmo Tong
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Weili Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Hongwei Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bin Liu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, H., Liu, B., Wang, X. et al. Total coloring of planar graphs without adjacent chordal 6-cycles. J Comb Optim 34, 257–265 (2017). https://doi.org/10.1007/s10878-016-0063-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10878-016-0063-3

Keywords

Search

Navigation