#### On dynamic coloring of certain cycle-related graphs

Arabian Journal of Mathematics
pp 1–9 | Cite as
On dynamic coloring of certain cycle-related graphs
AuthorsAuthors and affiliations
J. Vernold VivinN. MohanapriyaJohan KokM. Venkatachalam
Open Access
Article
First Online: 23 August 2018
Received: 31 January 2018
Accepted: 07 August 2018
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Abstract
Coloring the vertices of a particular graph has often been motivated by its utility to various applied fields and its mathematical interest. A dynamic coloring of a graph G is a proper coloring of the vertex set V(G) such that for each vertex of degree at least 2, its neighbors receive at least two distinct colors. A dynamic k-coloring of a graph is a dynamic coloring with k colors. A dynamic k-coloring is also called a conditional (k, 2)-coloring. The smallest integer k such that G has a dynamic k-coloring is called the dynamic chromatic number \(\chi _d(G)\) of G. In this paper, we investigate the dynamic chromatic number for the line graph of sunlet graph and middle graph, total graph and central graph of sunlet graphs, paths and cycles. Also, we find the dynamic chromatic number for Mycielskian of paths and cycles and the join graph of paths and cycles.
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Mathematics Subject Classification05C15
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References
1.
Ahadi, A.; Akbari, S.; Dehghan, A.; Ghanbari, M.: On the difference between chromatic number and dynamic chromatic number of graphs. Discret. Math. 312, 2579–2583 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
2.
Akbari, S.; Ghanbari, M.; Jahanbakam, S.: On the dynamic chromatic number of graphs. Combinatorics and Graphs. In: Contemporary Mathematics-American Mathematical Society, vol. 531, pp. 11–18 (2010)Google Scholar
3.
Akbari, S.; Ghanbari, M.; Jahanbekam, S.: On the list dynamic coloring of graphs. Discret. Appl. Math. 157, 3005–3007 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
4.
Alishahi, M.: Dynamic chromatic number of regular graphs. Discret. Appl. Math. 160, 2098–2103 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
5.
Bowler, N.; Erde, J.; Lehner, F.; Merker, M.; Pitz, M.; Stavropoulos, K.: A counterexample to Montgomery’s conjecture on dynamic colourings of regular graphs. Discret. Appl. Math. 229, 151–153 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
6.
Dehghan, A.; Ahadi, A.: Upper bounds for the \(2\)-hued chromatic number of graphs in terms of the independence number. Discret. Appl. Math. 160(15), 2142–2146 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
7.
Harary, F.: Graph Theory. Narosa Publishing home, New Delhi (1969)CrossRefzbMATHGoogle Scholar
8.
Kim, S.J.; Lee, S.J.; Park, W.J.: Dynamic coloring and list dynamic coloring of planar graphs. Discret. Appl. Math. 161, 2207–2212 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
9.
Lai, H.J.; Montgomery, B.; Poon, H.: Upper bounds of dynamic chromatic number. Ars Combinatoria 68, 193–201 (2003)MathSciNetzbMATHGoogle Scholar
10.
Li, X.; Zhou, W.: The \(2\)nd-order conditional \(3\)-coloring of claw-free graphs. Theor. Comput. Sci. 396, 151–157 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
11.
Li, X.; Yao, X.; Zhou, W.; Broersma, H.: Complexity of conditional colorability of graphs. Appl. Math. Lett. 22, 320–324 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
12.
Mohanapriya, N.; Vernold Vivin, J.; Venkatachalam, M.: \(\delta \)-Dynamic chromatic number of Helm graph families. Cogent Math. 3(1), 1178411, 1–4 (2016)Google Scholar
13.
Mohanapriya, N.; Vernold Vivin, J.; Venkatachalam, M.: On dynamic coloring of fan graphs. Int. J. Pure Appl. Math. 106, 169–174 (2016)Google Scholar
14.
Montgomery, B.: Dynamic coloring of graphs, ProQuest LLC, Ann Arbor, MI, Ph.D Thesis, West Virginia University (2001)Google Scholar
15.
Mycielski, J.: Surle coloriage des graphes. Colloquium Mathematicum 3, 161–162 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
16.
Taherkhani, A.: On \(r\)-dynamic chromatic number of graphs. Discret. Appl. Math. 201, 222–227 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
17.
Vernold Vivin, J.: Harmonious coloring of total graphs, \(n\)-leaf, central graphs and circumdetic graphs, Bharathiar University, (2007), Ph.D Thesis, Coimbatore, IndiaGoogle Scholar
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Authors and Affiliations
J. Vernold Vivin1Email authorView author's OrcID profileN. Mohanapriya2Johan Kok3M. Venkatachalam21.Department of MathematicsUniversity College of Engineering Nagercoil, (Anna University Constituent College)NagercoilIndia2.Department of MathematicsKongunadu Arts and Science CollegeCoimbatoreIndia3.Licensing ServicesMetro Police Head OfficeTshwaneSouth Africa