K. Kalaiselvi, N. Mohanapriya and J. Vernold Vivin

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 27, 2021, Number 2, Pages 191–200

DOI: 10.7546/nntdm.2021.27.2.191-200

**Full paper (PDF, 185 Kb)**

## Details

### Authors and affiliations

K. Kalaiselvi

*Department of Mathematics, Dr. Mahalingam College of Engineering and Technology
Pollachi-642 003, Tamil Nadu, India
*

N. Mohanapriya

*PG and Department of Mathematics, Kongunadu Arts and Science College
Coimbatore-641 029, Tamil Nadu, India
*

J. Vernold Vivin

*Department of Mathematics, University College of Engineering Nagercoil
(A Constituent College of Anna University, Chennai)
Konam, Nagercoil-629 004, Tamil Nadu, India
*

### Abstract

An -dynamic coloring of a graph is a proper coloring of such that every vertex in has neighbors in at least different color classes. The -dynamic chromatic number of graph denoted as , is the least such that has a coloring. In this paper we obtain the -dynamic chromatic number of the central graph, middle graph, total graph, line graph, para-line graph and sub-division graph of the comb graph denoted by , , , , and respectively by finding the upper bound and lower bound for the -dynamic chromatic number of the Comb graph.

### Keywords

*r*-dynamic coloring- Comb graph
- Central graph
- Middle graph
- Total graph
- Line graph
- Sub-division graph
- Para-line graph

### 2020 Mathematics Subject Classification

- 05C15
- 05C75

### References

- Adegoke, K. (2018). Partial sums and generating functions for power of second order sequences with indices in arithmetic progression. Preprint. Available online at: https://arxiv.org/abs/1904.09916.
- Belbachir, H., & Bencherif, F. (2013). Sums of products of generalized Fibonacci and Lucas numbers. Ars Combinatoria, 110, 33–43.
- Čerin, Z. (2015). Formulas for quadratic sums that involve generalized Fibonacci and Lucas numbers. Rad Hrvatske Akademije Znanosti i Umjetnosti, Matematicke Znanosti, 19(523), 1–12.
- Čerin, Z. (2009). On sum of products of Horadam numbers. Kyungpook Mathematical Journal, 49, 483–492.
- Čerin, Z. (2009) Sum of products of generalized Fibonacci and Lucas numbers.

Demonstratio Mathematica, 42, 247–258. - Horadam, A. F. (1965). Basic properties of a certain generalized sequence of numbers. The Fibonacci Quarterly, 3(3), 161–176.
- Khan, M. A., & Kwong, H. (2014). On sums of products of Fibonacci-type recurrences. The Fibonacci Quarterly, 52(2), 20–26.
- Kiliç, E., Ömür, N., & Ulutas, Y. T. (2011). Some finite sums involving generalized Fibonacci and Lucas numbers. Discrete Dynamics in Nature and Society, 2011(1), Article ID 284261.
- Kiliç, E., & Prodinger, H. (2015). Sums of products of generalized Fibonacci and Lucas numbers. Acta Mathematica Hungarica, 145(1), 17–25.
- Kiliç, E., & Stănică, P. (2013). General approach in computing sums of products of binary sequences. Hacettepe Journal of Mathematics and Statistics, 42(1), 1–7.
- Koshy, T. (2017). Fibonacci and Lucas Numbers with Applications, Wiley, New York.
- Koshy, T. (2014). Pell and Pell–Lucas Numbers with Applications, Springer, New York.
- Larcombe, P. J. (2017). Horadam sequences: A survey update and extension. Bulletin of the Institute of Combinatorics and its Applications, 80, 99–118.
- Ribenboim, P. (2000). My Numbers, My Friends, Springer, New York.
- Sloane, N. J. A. (ed.). The On-Line Encyclopedia of Integer Sequences. Available online at: https://oeis.org.
- Vajda, S. (2008). Fibonacci & Lucas Numbers, and the Golden Section: Theory and Applications, Dover, New York.

## Related papers

## Cite this paper

Kalaiselvi, K., Mohanapriya, N., & Vernold Vivin, J. (2021). On *r*-dynamic coloring of comb graphs. *Notes on Number Theory and Discrete Mathematics*, 27(2), 191-200, DOI: 10.7546/nntdm.2021.27.2.191-200.