On the total irregularity strength of convex polytope graphs

Authors

DOI:

https://doi.org/10.22199/issn.0717-6279-3959

Keywords:

Irregular assignment, Vertex irregular total k-labeling, Irregularity strength, Convex polytopes

Abstract

A vertex (edge) irregular total k-labeling ? of a graph G is a labeling of the vertices and edges of G with labels from the set {1,2,...,k} in such a way that any two different vertices (edges) have distinct weights. Here, the weight of a vertex x in G is the sum of the label of x and the labels of all edges incident with the vertex x, whereas the weight of an edge is the sum of label of the edge and the vertices incident to that edge. The minimum k for which the graph G has a vertex (edge) irregular total k-labeling is called the total vertex (edge) irregularity strength of G. In this paper, we are dealing with infinite classes of convex polytopes generated by prism graph and antiprism graph. We have determined the exact value of their total vertex irregularity strength and total edge irregularity strength.

Author Biographies

Syed Ahtshma Ul Haq Bokhary, Bahauddin Zakariya University.

Center for Advanced Studies in Pure and Applied Mathematics.

Muhammad Imran, United Arab Emirates University.

Department of Mathematical Sciences.

Usman Ali, Bahauddin Zakariya University.

Center for Advanced Studies in Pure and Applied Mathematics

 

References

A. Ahmad, “On vertex irregular total labelings of convex polytope graphs”, Utilitas Mathematica, vol. 89, pp. 69-78, 2012.

M. Bača, S. Jendrol, M. Miller. and J. Ryan, “On irregular total labellings”, Discrete Mathematics, vol. 307, pp. 1378-1388, 2007.

M. Bača, “Labellings of two classes of convex polytopes”, Utilitas Mathematica, vol. 34, pp. 24-31, 1988.

M. Bača, “On magic labellings of convex polytopes”, Annals of Discrete Mathematics, vol. 51, pp. 13-16, 1992.

G. S. Bloom and S. W. Golomb, “Applications of numbered undirected graphs”, Proceedings of the IEEE, vol. 65, no. 4, pp. 562-570, 1977.

G. S. Bloom and S. W. Golomb, “Numbered complete graphs, unusual rules, and assorted applications”, in Theory and Applications of Graphs, Y. Alavi and D. R. Lick, Eds. Berlin: Springer, 1978, pp. 53–65.

T. Bohman and D. Kravitz, “On the irregularity strength of trees”, Journal of Graph Theory, vol. 45, no. 4, pp. 241-254, 2004.

S. A. Bokhary, M. Imran, and A. Ahmad, “On vertex irregular total labeling of some cubic graphs”, Utilitas Mathematica, vol. 91, pp. 239-249, 2013.

G. Chartrand, M. S. Jacobson, J. Lehel, O. R. Oellermann, S. Ruiz, and F. Saba, “Irregular networks”, Congressus Numerantium, vol. 64, pp. 187-192, 1988.

R. J. Faudree, M. S. Jacobson, J. Lehel. and R. H. Schlep, “Irregular networks, regular graphs and integer matrices with distinct row and column sums”, Discrete Mathematics, vol. 76, no. 3, pp. 223-240, 1988.

A. Frieze, R. J. Gould, M. Karonski, and F. Pfender, “On graph irregularity strength”, Journal of Graph Theory, vol. 41, no. 2, pp. 120-137, 2002.

A. Gyárfás, “The irregularity strength of Km,m is 4 for odd m”, Discrete Mathematics, vol. 71, pp. 273-274, 1988.

M. Imran, S. A. Bokhary, and A. Q. Baig, “On families of convex polytopes with constant metric dimension”, Computers & Mathematics with Applications, vol. 60, pp. 2629-2638, 2010.

S. Jendrol, M. Tkáč and Z. Tuza, “The irregularity strength and cost of the union of cliques”, Discrete Mathematics, vol. 150, pp. 179-186, 1996.

E. Jucovič, Konvexné mnohosteny. Bratislava: Veda, 1981.

T. Nierhoff, “A tight bound on the irregularity strength of graphs”, SIAM Journal on Discrete Mathematics, vol. 13, no. 3, pp. 313-323, 2000.

J. Przybylo, “Linear bound on the irregularity strength and the total vertex irregularity strength of graphs”, SIAM Journal on Discrete Mathematics, vol. 23, no. 1, pp. 511-516, 2009.

K. Wijaya and Slamin, “Total vertex irregular labeling of wheels, fans, suns and friendship graphs”, Journal of Combinatorial Mathematics and Combinatorial Computing, vol. 65, pp. 103-112, 2008.

K. Wijaya, Slamin, S. Supangken and S. Jendrol, “Total vertex irregular labeling of complete bipartite graphs”, Journal of Combinatorial Mathematics and Combinatorial Computing, vol. 55, pp. 129-136, 2005.

Published

2021-06-23

How to Cite

[1]
S. A. U. H. Bokhary, M. Imran, and U. Ali, “On the total irregularity strength of convex polytope graphs”, Proyecciones (Antofagasta, On line), vol. 40, no. 5, pp. 1267-1277, Jun. 2021.

Issue

Section

Artículos