1. bookVolume 29 (2021): Issue 3 (November 2021)
Journal Details
License
Format
Journal
eISSN
1844-0835
First Published
17 May 2013
Publication timeframe
1 time per year
Languages
English
access type Open Access

On the r-dynamic coloring of some fan graph families

Published Online: 23 Nov 2021
Page range: 151 - 181
Received: 01 Feb 2021
Accepted: 15 May 2021
Journal Details
License
Format
Journal
eISSN
1844-0835
First Published
17 May 2013
Publication timeframe
1 time per year
Languages
English
Abstract

In this paper, we determine the r-dynamic chromatic number of the fan graph Fm,n and determine sharp bounds of this graph invariant for four related families of graphs: The middle graph M(Fm,n), the total graph T (Fm,n), the central graph C(Fm,n) and the line graph L(Fm,n). In addition, we determine the r-dynamic chromatic number of each one of these four families of graphs in case of being m = 1.

Keywords

MSC 2010

[1] Ahadi A, Akbari S, Dehghana A, Ghanbari M. On the difference between chromatic number and dynamic chromatic number of graphs. Discrete Mathematics 2012; 312 (17): 2579–2583.10.1016/j.disc.2011.09.006 Search in Google Scholar

[2] Akbari S, Ghanbari M, Jahanbakam S. On the dynamic chromatic number of graphs. In: Combinatorics and graphs. Contemporary Mathematics 2010; 531: 11–18.10.1090/conm/531/10454 Search in Google Scholar

[3] Alishahi M. On the dynamic coloring of graphs. Discrete Applied Mathematics 2011; 159 (2-3): 152–156.10.1016/j.dam.2010.10.012 Search in Google Scholar

[4] Alishahi M. Dynamic chromatic number of regular graphs. Discrete Applied Mathematics 2012; 160 (15): 2098–2103.10.1016/j.dam.2012.05.017 Search in Google Scholar

[5] Ansari N, Chandel RS, Jamal R. On b-chromatic number of prism graph families. Applications and Applied Mathematics. An International Journal (AAM) 2018; 13 (1): 286–295. Search in Google Scholar

[6] Deepa T, Falcón RM, Venkatachalam M. On the r-dynamic coloring of the direct product of a path with either a complete graph or a wheel graph. AIMS Mathematics 2021; 6 (2): 1470–1496.10.3934/math.2021090 Search in Google Scholar

[7] Deepa T, Venkatachalam M. On r-dynamic coloring of the total graphs of gear graphs. Applied Mathematics E-Notes 2018; 18: 69–81. Search in Google Scholar

[8] Deepa T, Venkatachalam M, Falcón RM. On the r-dynamic coloring of the direct product of a path with either a path or a cycle. AIMS Mathematics 2020; 5 (6): 6496–6520.10.3934/math.2020419 Search in Google Scholar

[9] Falcón RM, Venkatachalam M, Gowri S, Nandini G, On the r-dynamic coloring of the direct product of a path and a k-subdivision of a star graph. Discrete Mathematics, Algorithms and Applications 2021; online ready: DOI: 10.1142/S1793830921501214.10.1142/S1793830921501214 Search in Google Scholar

[10] Furmańczyk H, Vivin JV, Mohanapriya N. r-dynamic chromatic number of some line graphs. Indian Journal of Pure and Applied Mathematics 2018; 49: 591–600.10.1007/s13226-018-0288-1 Search in Google Scholar

[11] Gowri S, Venkatachalam M, Mishra VN, Mishra LN. On r-dynamic coloring of double star graph families. Palestine Journal of Mathematics 2021; 10: 53–62. Search in Google Scholar

[12] Hamada T, Yoshimura I. Traversability and connectivity of the middle graph of a graph. Discrete Mathematics 1976; 14 (3): 247–256.10.1016/0012-365X(76)90037-6 Search in Google Scholar

[13] Harary F. Graph Theory. Reading, Massachusetts: Addison Wesley, 1969.10.21236/AD0705364 Search in Google Scholar

[14] Jahanbekama S, Kim J, O S, West DB. On r-dynamic coloring of graphs. Discrete Applied Mathematics 2016; 206: 65–72.10.1016/j.dam.2016.01.016 Search in Google Scholar

[15] Kaliraj K, Kumar HN, Vivin JV. On dynamic colouring of Cartesian product of complete graph with some graphs. Journal of Taibah University for Science 2020; 14 (1): 168–171.10.1080/16583655.2020.1713586 Search in Google Scholar

[16] Kang R, Müller T, West DB. On r-dynamic coloring of grids. Discrete Applied Mathematics 2015; 186: 286–290.10.1016/j.dam.2015.01.020 Search in Google Scholar

[17] Kristiana AI, Utoyo MI, Alfarisi R, Dafik. r-dynamic coloring of the corona product of graphs. Discrete Mathematics, Algorithms and Applications 2020; 12 (2): 2050019.10.1142/S1793830920500196 Search in Google Scholar

[18] Lai HJ, Montgomery B, Poon H. Upper bounds of dynamic chromatic number. Ars Combinatoria 2003; 68: 193–201. Search in Google Scholar

[19] Lai HJ, Montgomery B, Shui T, Fan S. Conditional colorings of graphs. Discrete Mathematics 2006; 306 (16): 1997–2004.10.1016/j.disc.2006.03.052 Search in Google Scholar

[20] Loeb S, Mahoney T, Reiniger B, Wise J. Dynamic coloring parameters for graphs with given genus, Discrete Applied Mathematics 2019; 235: 129–141.10.1016/j.dam.2017.09.013 Search in Google Scholar

[21] Meganingtyas DEW. Analisis pewarnaan r-dinamis pada graf-graf khusus. PhD, Universitas Jember, UNEJ Repository, East Java, Indonesia, 2015. Search in Google Scholar

[22] Mohanapriya N, Vivin JV, Venkatachalam M. δ-dynamic chromatic number of helm graph families. Cogent Mathematics & Statistics 2016; 3 (1): 1178411.10.1080/23311835.2016.1178411 Search in Google Scholar

[23] Mohanapriya N, Vivin JV, Venkatachalam M. On dynamic coloring of fan graphs. International Journal of Pure and Applied Mathematics 2016; 106 (8): 169–174. Search in Google Scholar

[24] Montgomery B. Dynamic coloring of graphs. PhD Thesis, West Virginia University, ProQuest LLC, Ann Arbor, MI, 2001. Search in Google Scholar

[25] Nandini G, Venkatachalam M, Gowri S. On r-dynamic coloring of the family of bistar graphs. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 2019; 68 (1): 923–928. Search in Google Scholar

[26] Nandini G, Venkatachalam M, Falcón RM. On the r-dynamic coloring of subdivision-edge coronas of a path. AIMS Mathematics 2020; 5 (5): 4546–4562.10.3934/math.2020292 Search in Google Scholar

[27] Praveena K, Venkatachalam M. On equitable chromatic number of tad-pole graph Tm,n. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 2019; 68 (2): 1638–1646. Search in Google Scholar

[28] Rajalakshmi K, Venkatachalam M. On packing coloring of helm related graphs. Journal of Discrete Mathematical Sciences and Cryptography 2019: 22 (6): 989–1005.10.1080/09720529.2019.1640968 Search in Google Scholar

[29] Rohini A, Venkatachalam M. On irregular coloring of fan graph families. Journal of Physics: Conference Series 2018; 1139: 012061. Search in Google Scholar

[30] Rohini A, Venkatachalam M. On irregular coloring of wheel related graphs. Communications. Faculty of Sciences. University of Ankara. Series A1. Mathematics and Statistics 2019; 68 (2): 1462–1472. Search in Google Scholar

[31] Rohini A, Venkatachalam M. On irregular colorings of double wheel graph families. Communications. Faculty of Sciences. University of Ankara. Series A1. Mathematics and Statistics 2019; 68 (1): 944–949. Search in Google Scholar

[32] Roy S. Packing chromatic number of certain fan and wheel related graphs. AKCE International Journal of Graphs and Combinatorics 2017; 14 (1): 63–69.10.1016/j.akcej.2016.11.001 Search in Google Scholar

[33] Vivin JV, Venkatachalam M. A note on b-coloring of fan graphs. Journal of Discrete Mathematics Sciences and Cryptography 2014; 17 (5-6): 443–448.10.1080/09720529.2013.876781 Search in Google Scholar

Recommended articles from Trend MD

Plan your remote conference with Sciendo