On r- dynamic coloring of the family of bistar graphs

An r-dynamic coloring of a graph G is a proper coloring c of the vertices such that |c(N(v))|≥min{r,d(v)}, for each v∈V(G). The r-dynamic chromatic number of a graph G is the minimum k such that G has an r-dynamic coloring with k colors. In this paper, we obtain the r-dynamic chromatic number of middle, total, central and line graph of Bistar graph.

___

  • Bondy J.A. and Murty, U.S.R., Graph theory with applications, New York: Macmillan Ltd. Press, 1976.
  • Ahadi, A., Akbari, S., Dehghana, A. and Ghanbari, M., On the difference between chromatic number and dynamic chromatic number of graphs, Discrete Math. 312 (2012), 2579--2583.
  • Akbari, S., Ghanbari, M. and Jahanbakam, S., On the dynamic chromatic number of graphs, in: Combinatorics and Graphs, in: Contemp. Math.,(Amer. Math. Soc.,) 531 (2010), 11--18.
  • Akbari, S., Ghanbari, M. and Jahanbekam, S., On the list dynamic coloring of graphs, Discrete Appl. Math. 157 (2009), 3005--3007
  • Arundhadhi, R. and Ilayarani, V., Total coloring of closed helm, Flower and Bistar Graph Family, International journal of scientific and Research Publication, Vol 7, Issue 7, July 2017, ISSN 2250-3153.
  • Alishahi, M., Dynamic chromatic number of regular graphs, Discrete Appl. Math. 160 (2012), 2098--2103.
  • Dehghan, A. and Ahadi, A., Upper bounds for the 2-hued chromatic number of graphs in terms of the independence number, Discrete Appl. Math. 160(15) (2012), 2142--2146.
  • Harary, F., Graph Theory, Narosa Publishing home, New Delhi 1969.
  • Lai, H.J., Montgomery, B. and Poon, H., Upper bounds of dynamic chromatic number, Ars Combin. 68 (2003), 193--201.
  • Li, X. and Zhou, W., The 2nd-order conditional 3-coloring of claw-free graphs, Theoret. Comput. Sci. 396 (2008), 151--157.
  • Li, X. Yao, X., Zhou, W. and Broersma, H., Complexity of conditional colorability of graphs, Appl. Math. Lett. 22 (2009), 320--324.
  • Montgomery, B., Dynamic coloring of graphs, ProQuest LLC, Ann Arbor, MI, (2001), Ph.D Thesis, West Virginia University.
  • Taherkhani, A., r-Dynamic chromatic number of graphs, Discrete Appl. Math., 201(2016), 222--227.
  • Michalak, D., On middle and total graphs with coarseness number equal 1, Springer Verlag Graph Theory, Lagow proceedings, Berlin Heidelberg, New York, Tokyo, (1981), 139--150.
  • Vernold Vivin, J., Ph.D Thesis, Harmonious coloring of total graphs, n-leaf, central graphs and circumdetic graphs, Bharathiar University, (2007), Coimbatore, India
  • Arockia Aruldoss, J. and Pushparaj, S., Vertex Odd Mean and Even Mean Labeling of Fan Graph, Mongolian Tent, International Journal of Mathematics And its Applications, Volume 4, Issue 4 (2016), 223-227.