Some Problems in Total Graph Theory

  • M. Behzad
Part of the Mathematics and Its Applications book series (MAIA, volume 329)


Thirty years ago the total chromatic number and the total graph of a graph was introduced and a conjecture was stated in the author’s Ph.D. dissertation. This conjecture is known as the Total Chromatic Coiyecture (TCC). At this time numerous results concerning these and other total concepts such as total groups, total crossing numbers, and total Ramsey numbers exist in the literature. More specifically, during these decades results concerning different properties of total graphs, such as planarity, reconstructibility, and traversability have been obtained, and some parameters, including (vertex) connectivity, edge connectivity, arboricity, and eigenvalues, of such graphs have been studied. Some total concepts have been generalized, and some have been applied to other areas. In recent years a lot of attention has been paid to TCC and tremendous effort is being used toward its settlement. In this brief expository article we confine ourselves to some concepts and questions which are tangible even by some bright undergraduate students. As far as references are concerned, as usual, there are a lot of duplications. For the sake of briefness we have to be selective, and, with apology, we eliminate a lot of articles and results.


Chromatic Number Balance Incomplete Block Design Edge Connectivity Chromatic Polynomial Total Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    J. Akiyama and T. Hamada, The decomposition of the line graphs, middle graphs and total graphs, Discrete Math., 26 (1979), pp. 203–208.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    D. W. Bange, A. E. Barkauskas, and L. H. Host, Class-reconstruction of total graphs, J. Graph Theory, 11 (1987), pp. 221–230.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    M. Behzad, Graphs and their chromatic numbers, PhD dissertation, Michigan State University, Department of Mathematics, 1965.Google Scholar
  4. [4]
    M. Behzad, A criterion for the planarity of a total graph of a graph, Camb. Phil. Soc., 63 (1967), pp. 679–681.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    M. Behzad, A characterization of total graphs, Proc. Amer. Math. Soc., 26 (1970), pp. 383–389.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    M. Behzad, The total chromatic number of a graph, a survey, in Comb. Math. and its Applications, D. J. A. Welsh, ed., Acad. Press, 1971, pp. 1–9.Google Scholar
  7. [7]
    M. Behzad and G. Chartrand, Total graphs and traversability, Proc. Edinburgh Math. Soc., 15 Ser. II (1966), pp. 117–120.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    M. Behzad, An introduction to total graphs, in Proc. of the Inter. Symposium on Theory of Graphs, Rome, 1967, pp. 31–33.Google Scholar
  9. [9]
    M. Behzad, G. Chartrand, and J. K. Cooper Jr., The colour numbers of complete graphs, J. London Math. Soc., 42 (1967), pp. 226–228.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    M. Behzad, G. Chartrand, and L. Lesniak-Foster, Graphs and Digraphs, Wadsworth International Group, Belmont,California, 1979.Google Scholar
  11. [11]
    M. Behzad and E. S. Mahmoodian, Graphs versus designs: A quasisurvey, in Graph Theory, Combinatorics, and Applications, Y. Alavi and et al, eds., John Wiley and Sons, Inc., 1991, pp. 125–141.Google Scholar
  12. [12]
    M. Behzad, Eccentric sequences and triangle sequences of block designs, Discrete Math., 127 (1994), pp. 47–56.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    M. Behzad and H. Radjavi, The total group of a graph, Proc. American Math. Soc., 19 (1968), pp. 158–163.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    M. Behzad, Structure of regular total graphs, J. London Math. Soc., 44 (1969), pp. 433–436.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    M. Behzad, Another analog of Ramsey numbers, Math. Ann., 186 (1970), pp. 228–232.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    N. Biggs, Algebraic Graph Theory, Camb. Univ. Press, 1974.zbMATHGoogle Scholar
  17. [17]
    B. Bollobás and A. J. Harris, List-colourings of graphs, Graphs and Comb., 1 (1985), pp. 115–127.zbMATHCrossRefGoogle Scholar
  18. [18]
    J. A. Bondy, A graph reconstructor’s manual, London Math. Soc. Lecture Note Series, 166 (1991), pp. 221–252.MathSciNetGoogle Scholar
  19. [19]
    G. Chartrand and M. J. Stewart, Total digraphs, Canadian Math. Bull., 9 (1966), pp. 171–176.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    A. G. Chetwynd, Total colorings of graphs-a progress report, in Graph Colorings, R. Nelson and R. J. Wilson, eds., Research Notes in Mathematics, Pitman, 1990, pp. 65–77.Google Scholar
  21. [21]
    A. G. Chetwynd and R. Häggkvist, Some upper bounds on the total and list chromatic numbers of multigraphs, J. Graph Theory, 16 (1992), pp. 503–516.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    A. G. Chetwynd and A. J. W. Hilton, Some refinements of the total chromatic number conjecture, Congressus Numerantium, 66 (1988), pp. 195–216.MathSciNetGoogle Scholar
  23. [23]
    A. G. Chetwynd, A. J. W. Hilton, and Zhao Cheng, The total chromatic number of graphs of high minimum degree, J. London Math. Soc. (2), 44 (1991), pp. 193–202.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    D. M. Cvetković, Spectrum of the total graph of a graph, Publications de L’Ins tit lit Mathematique, 16 (1973), pp. 49–52.zbMATHGoogle Scholar
  25. [25]
    F. Gavril, A recognition algorithm for the total graphs, Networks, 8 (1978), pp. 121–133.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    R. L. Graham, B. L. Rothschild, and J. H. Spencer, Ramsey Theory, John Wiley & Sons, 1990.zbMATHGoogle Scholar
  27. [27]
    A. J. W. Hilton, Recent results on the total chromatic number, Discrete Math., III (1993), pp. 323–331.Google Scholar
  28. [28]
    A. J. W. Hilton and H. R. Hind, The total chromatic number of graphs having large maximum degree, Discrete Math., 117 (1993), pp. 127–140.MathSciNetzbMATHCrossRefGoogle Scholar
  29. [29]
    H. R. Hind, An upper bound for the total chromatic number, Graphs and Comb., 6 (1990), pp. 153–159.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    H. R. Hind, A summary of total coloring results. Preprint, May 29 1991.Google Scholar
  31. [31]
    H. R. Hind, An upper bound for the total chromatic number of dense graphs, J. Graph Theory, 16 (1992), pp. 197–203.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    H. R. Hind, Recent developments in total coloring, Discrete Math., 125 (1994), pp. 211–218.MathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    A. V. Kostochka, The total colouring of a multigraph with maximal degree 4, Discrete Math., 17 (1977), pp. 161–163.MathSciNetzbMATHCrossRefGoogle Scholar
  34. [34]
    A. V. Kostochka, An analogue of Shannon’s estimate for complete colorings (Russian), Diskret. Analiz. 30 (1977), pp. 13–22.zbMATHGoogle Scholar
  35. [35]
    I. Krasikŏv and Y. Roditty, Recent applications of Nash-Williams lemma to the edge-reconstruction conjecture, Ars Comb., Ser. A, 29 (1990), pp. 215–224.Google Scholar
  36. [36]
    V. R. Kulli and N. S. Annigeri, Total graphs with crossing number 1, J. Math. Phys. Sci., 12 (1978), pp. 615–617.MathSciNetzbMATHGoogle Scholar
  37. [37]
    Van Bang Lê, Perfect k-line graphs and k-total graphs, J. Graph Theory, 17 (1993), pp. 65–73.MathSciNetzbMATHCrossRefGoogle Scholar
  38. [38]
    C. J. H. McDiarmid and B. Reed, On total colourings of graphs, J. Comb. Theory, Ser. B, 57 (1993), pp. 122–130.MathSciNetzbMATHCrossRefGoogle Scholar
  39. [39]
    C. J. H. McDiarmid and A. Sanchéz-Arroyo, An upper bound for total colouring of graphs. Discrete Math., III (1993), pp. 389–392.Google Scholar
  40. [40]
    J. C. Meyer, Nombre chromatique total d’un hypergraphe, J. Comb. Theory, Ser. B, 24 (1978), pp. 44–50.zbMATHCrossRefGoogle Scholar
  41. [41]
    C. St. J. A. Nash-Williams, The reconstruction problem, in Selected Topics in the Theory of Graphs, L. W. Beineke and R. J. Wilson, eds., Academic Press, 1978.Google Scholar
  42. [42]
    N. P. Patil, Forbidden subgraphs and total graphs with crossing number 1, J. Math. Phys. Sci., 17 (1983), pp. 293–295.MathSciNetzbMATHGoogle Scholar
  43. [43]
    S. B. Rao and G. Ravindra, A characterization of perfect total graphs, J. Math phys. Sci., 11 (1977), pp. 25–26.MathSciNetzbMATHGoogle Scholar
  44. [44]
    M. Rosenfeld, On the total chromatic number of certain graphs, Israel J. Math., 9 (1971), pp. 396–402.MathSciNetzbMATHCrossRefGoogle Scholar
  45. [45]
    A. Saito and Tian Songlin, The binding number of line graphs and total graphs, Graphs and Comb., 1 (1985), pp. 351–356.MathSciNetzbMATHCrossRefGoogle Scholar
  46. [46]
    N. Vijayaditya, On total chromatic number of a graph, J. London Math. Soc., 3 (1971), pp. 405–408.MathSciNetzbMATHCrossRefGoogle Scholar
  47. [47]
    H. P. Yap, On the total chromatic number of a graph, Research Report No. 343, 1988.Google Scholar
  48. [48]
    H. P. Yap, Wang Jian-Fang, and Zhang Zhongfu, Total chromatic number of graphs of high degree, J. Aust. Math. Soc., Ser. A, 47 (1989), pp. 445–452.zbMATHCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • M. Behzad
    • 1
  1. 1.Beheshti UniversityTehranIran

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