ArticleOriginal scientific text
Title
Total vertex irregularity strength of disjoint union of Helm graphs
Authors 1, 2, 3
Affiliations
- College of Computer Science and Information Systems, Jazan University, Jazan, Kingdom of Saudi Arabia
- Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Indonesia
- Center for Advanced Mathematics and Physics (CAMP), National University of Science and Technology (NUST), H-12 Sector, Islamabad, Pakistan
Abstract
A total vertex irregular k-labeling φ 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 for any two different vertices x and y their weights wt(x) and wt(y) are distinct. 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. The minimum k for which the graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G. We have determined an exact value of the total vertex irregularity strength of disjoint union of Helm graphs.
Keywords
vertex irregular total k-labeling, Helm graphs, total vertex irregularity strength
Bibliography
- A. Ahmad and M. Bača, On vertex irregular total labelings, Ars Combin. (to appear).
- A. Ahmad, K.M. Awan, I. Javaid, and Slamin, Total vertex irregularity strength of wheel related graphs, Australas. J. Combin. 51 (2011) 147-156.
- M. Anholcer, M. Kalkowski and J. Przybyło, A new upper bound for the total vertex irregularity strength of graphs, Discrete Math. 309 (2009) 6316-6317, doi: 10.1016/j.disc.2009.05.023.
- M. Bača, S. Jendrol', M. Miller and J. Ryan, On irregular total labellings, Discrete Math. 307 (2007) 1378-1388, doi: 10.1016/j.disc.2005.11.075.
- T. Bohman and D. Kravitz, On the irregularity strength of trees, J. Graph Theory 45 (2004) 241-254, doi: 10.1002/jgt.10158.
- G. Chartrand, M.S. Jacobson, J. Lehel, O.R. Oellermann, S. Ruiz and F. Saba, Irregular networks, Congr. Numer. 64 (1988) 187-192.
- 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 Math. 76 (1988) 223-240, doi: 10.1016/0012-365X(89)90321-X.
- A. Frieze, R.J. Gould, M. Karoński, and F. Pfender, On graph irregularity strength, J. Graph Theory 41 (2002) 120-137, doi: 10.1002/jgt.10056.
- A. Gyárfás, The irregularity strength of
- S. Jendrol', M. Tkáč and Zs. Tuza, The irregularity strength and cost of the union of cliques, Discrete Math. 150 (1996) 179-186, doi: 10.1016/0012-365X(95)00186-Z.
- M. Kalkowski, M. Karoński and F. Pfender, A new upper bound for the irregularity strength of graphs, SIAM J. Discrete Math. 25 (2011) 139-1321, doi: 10.1137/090774112.
- T. Nierhoff, A tight bound on the irregularity strength of graphs, SIAM J. Discrete Math. 13 (2000) 313-323, doi: 10.1137/S0895480196314291.
- Nurdin, E.T. Baskoro, A.N.M. Salamn and N.N. Goas, On the total vertex irregularity strength of trees, Discrete Math. 310 (2010) 3043-3048, doi: 10.1016/j.disc.2010.06.041.
- J. Przybyło, Linear bound on the irregularity strength and the total vertex irregularity strength of graphs, SIAM J. Discrete Math. 23 (2009) 511-516, doi: 10.1137/070707385.
- K. Wijaya and Slamin, Total vertex irregular labeling of wheels, fans, suns and friendship graphs, J. Combin. Math. Combin. Comput. 65 (2008) 103-112.
- K. Wijaya, Slamin, Surahmat and S. Jendrol, Total vertex irregular labeling of complete bipartite graphs, J. Combin. Math. Combin. Comput. 55 (2005) 129-136.
Pages:
427-434
Main language of publication
English
Received
2011-04-12
Accepted
2011-07-20
Published
2012
DOI: 10.7151/dmgt.1619
Exact and natural sciences