Totally antimagic total labeling of helm and gear graphs
Abstract
A total labeling of a graph G is a bijection from the union of the vertex set and the edge set of G to the set {1,2,...,|V(G)|+|E(G)|}. Under a total labeling, the vertex-weight of a vertex is defined as the sum of its label and the labels of all edges incident to it. Similarly, the edge-weight of an edge is the sum of its label and the labels of its two end vertices. A total labeling is said to be edge-antimagic total if all the edge-weights are pairwise distinct, and vertex-antimagic total if all the vertex-weights are pairwise distinct. If a total labeling is edge-antimagic total and vertex-antimagic total at the same time, then it is called a totally antimagic total labeling. A graph that admits totally antimagic total labeling is called a totally antimagic total graph. In this paper, we show that helm graphs Hn and gear graphs Gn are totally antimagic total graphs.
Keywords
Full Text:
PDFDOI: http://dx.doi.org/10.19184/ijc.2026.10.1.3
References
M. A. Ahmed and J. B. Babujee, Totally antimagic total labeling of complete bipartite graphs, Rom. J. Math. Comput. Sci., 7(2017), 21-28.
J. A. Gallian, A Dynamic Survey of Graph Labeling, Electron. J. Combin., Dynamic Survey DS6, 2018.
M. Baca, M. Miller, O. Phanalasy, J. Ryan, A. Semanicova-Fenovcikova and A. A. Sillasen, Totally antimagic total graphs, Australas. J. Combin., 61(2015), 42-56.
M. Baca, J. MacDougall, F. Bertault, M. Miller, R. Simanjuntak and Slamin, Vertex-antimagic total labelings of graphs, Discuss. Math. Graph Theory, 23(2003), 67-83.
J. Clark and D. A. Holton, A First Look at Graph Theory, World Scientific, Singapore, 1991.
N. Hartsfield and G. Ringel, Pearls in Graph Theory, Academic Press, Boston, 1990.
A. Solairaju and M. A. Arockiasamy, Gracefulness of k-step staircase graphs, J. Anal. Comput., 6(2010), 109-114.
J.-L. Shang, C. Lin and S.-C. Liaw, On the antimagic labeling of star forests, Util. Math., 97(2015), 373-385.
N. Alon, G. Kaplan, A. Lev, Y. Roditty and R. Yuster, Dense graphs are antimagic, J. Graph Theory, 47(2004), 297-309.
R. Simanjuntak, F. Bertault and M. Miller, Two new (a,d)-antimagic graph labelings, In: Proc. Eleventh Australasian Workshop on Combinatorial Algorithms, 11(2000), 179-189.
S. Arumugam and M. Nalliah, Super (a,d)-edge antimagic total labelings of friendship graphs, Australas. J. Combin., 53(2012), 237-244.
M. A. Ahmed, J. Baskar Babujee, M. Baca and A. Semanicova-Fenovcikova, On totally antimagic, edge-magic and vertex-antimagic total graphs, Utilitas Math., 111(2019), 161-173.
G. Ali, M. Baca, Y. Lin and A. Semanicova-Fenovcikova,
Super-vertex-antimagic total labelings of disconnected graphs,
Discrete Math., 309(2009), 6048-6054.
Y. Lin, M. Baca, M. Miller and K. Sugeng, Super (a,d)-vertex-antimagic total labelings, J. Combin. Math. Combin. Comput., 55(2005), 91-102.
A. Rosa, On certain valuations of the vertices of a graph,
in: Theory of Graphs (Internat. Symposium, Rome, July 1966), Gordon and Breach, New York, and Dunod, Paris, 1967, pp.349-355.
Refbacks
- There are currently no refbacks.
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.