Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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COMPUTATION OF TOTAL CHROMATIC NUMBER FOR CERTAIN CONVEX POLYTOPE GRAPHS

  • A. PUNITHA (Department of Mathematics, Vels Institute of Science, Technology and Advanced Studies(VISTAS)) ;
  • G. JAYARAMAN (Department of Mathematics, Vels Institute of Science, Technology and Advanced Studies(VISTAS))
  • Received : 2023.09.24
  • Accepted : 2023.12.18
  • Published : 2024.05.30
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Abstract

A total coloring of a graph G is an assignment of colors to the elements of a graphs G such that no adjacent vertices and edges receive the same color. The total chromatic number of a graph G , denoted by χ''(G), is the minimum number of colors that suffice in a total coloring. In this paper, we proved the Behzad and Vizing conjecture for certain convex polytope graphs Dpn, Qpn, Rpn, En, Sn, Gn, Tn, Un, Cn,respectively. This significant result in a graph G contributes to the advancement of graph theory and combinatorics by further confirming the conjecture's applicability to specific classes of graphs. The presented proof of the Behzad and Vizing conjecture for certain convex polytope graphs not only provides theoretical insights into the structural properties of graphs but also has practical implications. Overall, this paper contributes to the advancement of graph theory and combinatorics by confirming the validity of the Behzad and Vizing conjecture in a graph G and establishing its relevance to applied problems in sciences and engineering.

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References

  1. L.J. Aleksandar, Savic, L.J. Zoran, Maksimovic, Milena S. Bogdanovic, The open-locating-dominating number of some convex polytopes, Faculty of sciences and mathematics 32 (2018), 635-642. https://doi.org/10.2298/FIL1802635S
  2. M. Bezhad, Graphs and their chromatic numbers, Doctoral Thesis, Michigan State University, 1965.
  3. M. Behzad, G. Chartrand and J.K. Cooper, The color numbers of complete graphs, Journal London Math. Soc. 42 (1967), 226-228. https://doi.org/10.1112/jlms/s1-42.1.226
  4. O.V. Borodin, On the total coloring planar graphs, J. Reine Angew Math. 394 (1989), 180-185. https://doi.org/10.1515/crll.1989.394.180
  5. M. Imran, U.H. Bokhary, A.Q. Baig, On the metric dimension of rotationally-symmetric convex polytopes, Journal of Algebra Combinatorics Discrete Structures and Applications 3 (2015), 45-59. https://doi.org/10.13069/jacodesmath.47485
  6. A.V. Kostochka, The total coloring of a multigraph with maximal degree 4, Discrete Math. 17 (1989), 161-163. https://doi.org/10.1016/0012-365X(77)90146-7
  7. A.V. Kostochka, The total chromatic number of any multigraph with maximum degree five is at most seven, Discrete Math. 162 (1996), 199-214. https://doi.org/10.1016/0012-365X(95)00286-6
  8. Martin Baca, Labellings of two classes of convex polytopes, Utilitas Mathametica 34 (1988), 24-31.
  9. Martin Baca, On magic labelling of convex polytopes, Annals of Discrete mathematics 51 (1992), 13-16. https://doi.org/10.1016/S0167-5060(08)70599-5
  10. K. Manikandan, T. Harikrishnan, Equitable coloring of some convex polytope graphs, International Journal of Applied and computational mathematics 4 (2018), 1-8. https://doi.org/10.1007/s40819-018-0554-2
  11. D. Muthuramakrishan and G. Jayaraman, Total coloring of certain graphs, Advances and applications in discrete mathematics 27 (2021), 31-38. https://doi.org/10.17654/DM027010031
  12. Muhammad Imran, Syed Ahtsham U.I. Haq Bokhary, A.Q. Baig, On families of convex polytopes with constant metric dimensions, Computers and Mathematics with Applications 60 (2010), 2629-2638. https://doi.org/10.1016/j.camwa.2010.08.090
  13. M. Rosanfeld, On the total colouring of certain graphs, Israel J. Math. 9 (1972), 396-402. https://doi.org/10.1007/BF02771690
  14. N. Vijayaditya, On total chromatic number of a graph, J. London Math Soc. (1971), 405- 408.
  15. V.G. Vizing, Some unsolved problems in graph theory, Russian Mathematical Survey 23 (1968), 125-141. https://doi.org/10.1070/RM1968v023n06ABEH001252
  16. H.P. Yap, Total colourings of graphs, Lecture Notes in Mathematics, Springer, Berlin, 1623, 1996.
  17. H.P. Yap and K.H. Chew, The chromatic number of graphs of high degree, II, J. Austral. Math. Soc. (Series-A) 47 (1989), 445-452. https://doi.org/10.1017/S1446788700033176
  18. Yu-Ming Chu,Muhammad Faisal Nadeem, Muhammad Azeem and Muhammad Kamran Siddiqui, On sharp bounds on partition dimension of convex polytopes, IEEE Access. DOI 10.1109/Access.2020.3044498