Science in China Series A: Mathematics

, Volume 52, Issue 5, pp 973–980 | Cite as

Adjacent strong edge colorings and total colorings of regular graphs

  • ZhongFu Zhang
  • Douglas R. Woodall
  • Bing Yao
  • JingWen Li
  • XiangEn Chen
  • Liang Bian


It is conjectured that X as (G) = X t (G) for every k-regular graph G with no C 5 component (k ⩾ 2). This conjecture is shown to be true for many classes of graphs, including: graphs of type 1; 2-regular, 3-regular and (|V(G)| - 2)-regular graphs; bipartite graphs; balanced complete multipartite graphs; k-cubes; and joins of two matchings or cycles.


graph total coloring adjacent strong edge coloring 


05C15 68R10 


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  1. 1.
    Zhang Z F, Liu L Z, Wang J F. Adjacent strong edge coloring of graphs. Appl Math Lett, 15: 623–626 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Behzad M. Graphs and their chromatic numbers. PhD thesis. Michigan State University, 1965Google Scholar
  3. 3.
    Behzad M. The total chromatic number of a graph: A survey. In: Welsh D J A, eds. Combinatorial Mathematics and its Applications. Proc Conf, Oxford, 1969. London: Academic Press, 1971, 1–8Google Scholar
  4. 4.
    Vizing V G. On an estimate of the chromatic class of a p-graph (in Russian). Metody Diskret Analiz, 3: 25–30 (1964)MathSciNetGoogle Scholar
  5. 5.
    Vizing V G. The chromatic class of a multigraph (in Russian). Kibernetika, 3: 29–39 (1965)MathSciNetGoogle Scholar
  6. 6.
    Zhang Z F, Zhang J X, Wang J W. The total chromatic number of some graphs. Sci China Ser A, 31: 1434–1441 (1988)zbMATHGoogle Scholar
  7. 7.
    Zhang Z F, Chen X E, Li J W, et al. On adjacent-vertex-distinguishing total coloring of graphs. Sci China Ser A, 48(3): 289–299 (2005)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Zhang Z F, Li J W, Chen X E, et al. D(β)-vertex-distinguishing total coloring of graphs. Sci China Ser A, 49(10): 1430–1440 (2006)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Zhang Z F, Cheng H, Yao B, et al. On the adjacent-vertex-strongly-distinguishing total coloring of graphs. Sci China Ser A, 51(3): 427–436 (2008)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Zhang Z F, Qiu P X, Xu B G, et al. Vertex-distinguishing total coloring of graphs. Ars Comb, 87: 33–45 (2008)MathSciNetGoogle Scholar
  11. 11.
    Zhang Z F, Wang J F, Wang W F, et al. The complete chromatic number of some planar graphs. Sci China Ser A-Math, 36: 1169–1177 (1993)zbMATHGoogle Scholar
  12. 12.
    Rosenfeld M. On the total coloring of certain graphs. Israel J Math, 9: 396–402 (1971)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Vijayaditya N. On the total chromatic number of a graph. J London Math Soc, 3(2): 405–408 (1971)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Balister P N, Györi E, Lehel J, et al. Adjacent vertex distinguishing edge-colorings. SIAM J Discrete Math, 21(1): 237–250 (2007)CrossRefGoogle Scholar
  15. 15.
    König D. Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre. Math Ann, 77: 453–465 (1916)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Bermond J C. Nombre chromatique total du graphe r-parti complet. J London Math Soc, 9(2): 279–285 (1974)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Černý J, Horňák M, Soták R. Observability of a graph. Math Slovaca, 46: 21–31 (1996)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Horňák M, Soták R. Observability of complete multipartite graphs with equipotent parts. Ars Combin, 41: 289–301 (1995)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Zhao Q C, Zhang Z F. On the total chromatic number of k-cubes. J Taiyuan Institute of Machinery, 14: 292–296 (1993)Google Scholar

Copyright information

© Science in China Press and Springer-Verlag GmbH 2009

Authors and Affiliations

  • ZhongFu Zhang
    • 1
    • 2
  • Douglas R. Woodall
    • 3
  • Bing Yao
    • 2
  • JingWen Li
    • 1
  • XiangEn Chen
    • 2
  • Liang Bian
    • 4
  1. 1.Institute of Applied MathematicsLanzhou Jiaotong UniversityLanzhouChina
  2. 2.College of Mathematics and Information ScienceNorthwest Normal UniversityLanzhouChina
  3. 3.School of Mathematical SciencesUniversity of NottinghamNottinghamUK
  4. 4.School of Mathematical SciencesQufu Normal UniversityQufuChina

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