On the r-dynamic coloring of some fan graph families

Falcón, Raúl M.; Venkatachalam, M.; Gowri, S.; Nandini, G.

doi:10.2478/auom-2021-0039

Abstract

In this paper, we determine the r-dynamic chromatic number of the fan graph F_m,n and determine sharp bounds of this graph invariant for four related families of graphs: The middle graph M(F_m,n), the total graph T (F_m,n), the central graph C(F_m,n) and the line graph L(F_m,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.

References

[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 Back to article
[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 Back to article
[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 Back to article
[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 Back to article
[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 Back to article
[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 Back to article
[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 Back to article
[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 Back to article
[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 Back to article
[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 Back to article
[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 Back to article
[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 Back to article
[13] Harary F. Graph Theory. Reading, Massachusetts: Addison Wesley, 1969.10.21236/AD0705364
Search in Google Scholar Back to article
[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 Back to article
[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 Back to article
[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 Back to article
[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 Back to article
[18] Lai HJ, Montgomery B, Poon H. Upper bounds of dynamic chromatic number. Ars Combinatoria 2003; 68: 193–201.
Search in Google Scholar Back to article
[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 Back to article
[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 Back to article
[21] Meganingtyas DEW. Analisis pewarnaan r-dinamis pada graf-graf khusus. PhD, Universitas Jember, UNEJ Repository, East Java, Indonesia, 2015.
Search in Google Scholar Back to article
[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 Back to article
[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 Back to article
[24] Montgomery B. Dynamic coloring of graphs. PhD Thesis, West Virginia University, ProQuest LLC, Ann Arbor, MI, 2001.
Search in Google Scholar Back to article
[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): 923–928.10.31801/cfsuasmas.489727
Search in Google Scholar Back to article
[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 Back to article
[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): 1638–1646.10.31801/cfsuasmas.546904
Search in Google Scholar Back to article
[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 Back to article
[29] Rohini A, Venkatachalam M. On irregular coloring of fan graph families. Journal of Physics: Conference Series 2018; 1139: 012061.
Search in Google Scholar Back to article
[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 Back to article
[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 Back to article
[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 Back to article
[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 Back to article

On the r-dynamic coloring of some fan graph families

Abstract

Paradigm

My account