TARU PUBLICATIONS
Journal of Discrete Mathematical Sciences and Cryptography cover
Hybrid ·Peer-reviewed·ISSN (Online): 2169-0065·ISSN (Print): 0972-0529

Monthly Journal: Publishes theoretical and applied research in all areas of Discrete Mathematical Sciences, Cryptography, Combinatorics, Elliptic Curves and Information Security.

Issues up to 2022 co-published with and available at:Taylor & Francis Online
submissions@tarupublications.com
Open Access Research Article

A characterization of some types of Cayley graphs and addition Cayley graphs and their total chromatic numbers

, *

* Corresponding author · click or hover a name for details

pp. 963–981Vol. 27Issue 3April 2024DOI: 10.47974/JDMSC-1621 Crossmark XML
Received:
11 Aug 2021
Published Online:
01 May 2024
Article type:
Research Article
Language:
EN
Article no.:
JDMSC-1621
Pages:
963–981

Abstract

Let ℤn  be the set of integer modulo n. The Cayley graph on ℤn is an undirected graph whose vertex set is ℤn and two vertices a, b are adjacent if and only if a – b ϵ S ⸦  ℤn\{0}. The addition Cayley graph on ℤn  is a graph whose vertex set is ℤn  and two vertices a, b are adjacent if and only if a + b ϵ A ⸦ ℤn.  In this paper , we characterize Cayley graphs and addition Cayley graphs of even orders. Their basic properties of them are investigated. We also give exact values for the total chromatic numbers of Cayley graphs and addition Cayley graphs where their orders are even integers. Moreover, we provide examples to illustrate these results.

Keywords

Subject Classifications

05C1505C25

References

[1] M. Behzad, G. Chartrand, and J.K. Cooper Jr, The Colour Numbers of Complete Graphs, J. London Math. Soc., Vol. 42 (1967), pp. 226–228.
[2] M. Behzad, Graphs and Their Chromatic Numbers, Doctoral Thesis, Michigan State University, 1965.
[3] N. Biggs, N.L. Biggs, and B. Norman, Algebraic Graph Theory, Cambridge university press, 1993.
[4] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications, Macmillan, London and Elsevier, New York, 1976.
[5] O.V. Borodin, On the Total Coloring of Planar Graphs, J. Reine Angew. Math., Vol. 394 (1989), pp. 180–185.
[6] R.L. Brooks, On Colouring the Nodes of a Network, Proc. Cambridge Philos. Soc., Vol. 37(2) (1941), pp. 194–197.
[7] K. Chew and H. Yap, Total Chromatic Number of Complete r-partite Graphs, J. Graph Theory 16 (1992), pp. 629–634.
[8] B. Cheyne, V. Gupta, and C. Wheeler, Hamilton Cycles in Addition Araphs, Rose-Hulman Undergrad. Math J., Vol. 4(1) (2003), p. 6.
[9] N. Ghanbari and S. Alikhani, More on the Total Dominator Chromatic Number of a Graph, J. Inf. Optim. Sci., Vol. 40(1) (2019), pp. 157–169.
[10] C. Godsil and G.F. Royle, Algebraic Graph Theory, Vol. 207, Springer-Verlag, New York, 2001.
[11] D. Grynkiewicz, V.F. Lev, and O. Serra, The Connectivity of Addition Cayley Graphs, Electron. Notes Discrete Math., Vol. 29 (2007), pp. 135–139.
[12] D. Grynkiewicz, V.F. Lev, and O. Serra, Connectivity of Addition Cayley Graphs, J. Combin. Theory Ser. B., Vol. 99(1) (2009), pp. 202–217.
[13] A. Gupta, J. Geetha, and K. Somasundaram, Total Coloring Algorithm for Graphs, Appl. Math. Sci. 9 (2015), pp. 1297–1302.
[14] D. König, Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre, Math. Ann., Vol. 77(4) (1916), pp. 453–465.
[15] A.V. Kostochka, The Total Coloring of a Multigraph with Maximal Degree 4, Discrete Math., Vol. 17(2) (1977), pp. 161–163.
[16] M.E. Leidner, A Study of the Total Coloring of Graphs, Electronic Theses and Dissertations. University of Louisville, 2012.
[17] C. Promsakon, Some Properties of Addition Cayley Graphs, MJ-MATh., Vol. 62(693) (2017), pp. 17–24.
[18] A.V. Rani and N. Parvathi, Chromatic Number of Some Families of Graphs, J. Discret. Math. Sci. Cryptogr., Vol. 22(6) (2019), pp. 1141–1149.
[19] M. Rosenfeld, On the Total Coloring of Certain Graphs, Isr. J. Math., Vol. 9(3) (1971), pp. 396–402.
[20] D. Sinha, P. Garg, and A. Singh, Some Properties of Unitary Addition Cayley Graphs, Notes Number Theory Discrete Math., Vol. 17(3) (2011), pp. 49–59.
[21] T. Srinivasa Murthy, A Proof of the Total Coloring Conjecture, arXiv preprint arXiv:2003.09658 (2020).
[22] N. Vijayaditya, On Total Chromatic Number of a Graph, J. London Math. Soc., Vol. 2(3) (1971), pp. 405–408.
[23] V.G. Vizing, On an Estimate of the Chromatic Class of a p-Graph, Discret. Analiz., Vol. 3 (1964), pp. 25–30.
[24] R.J. Wilson, History of Graph Theory, in Handbook of Graph Theory, CRC Press (2013). Available at https://www.routledgehandbooks.com/doi/10.1201/b16132-3.
[25] H.P. Yap, Total Colourings of Graphs, Lecture Notes in Mathematics, Vol. 1623, Springer, Berlin, 1996.

Views: 472Downloads: 62Citations: 0