### On the total vertex irregularity strength of comb product of two cycles and two stars

Rismawati Ramdani

#### Abstract

Let = (V(G),E(G)) be a graph and k be a positive integer. A total k-labeling of G is a map f : V ∪ E → {1,2,3,...,k}. The vertex weight v under the labeling f is denoted by w_f(v) and defined by w_f(v) = f(v) + \sum_{uv \in{E(G)}} {f(uv)}. A total k-labeling of G is called vertex irregular if there are no two vertices with the same weight. The total vertex irregularity strength of G, denoted by tvs(G), is the minimum k such that G has a vertex irregular total k-labeling. This labelings were introduced by Baca, Jendrol, Miller, and Ryan in 2007. Let G and H be two connected graphs. Let o be a vertex of H. The comb product between G and H, denoted by \rhd_o H, is a graph obtained by taking one copy of G and |V(G)| copies of H and grafting the i-th copy of H at the vertex o to the i-th vertex of G. In this paper, we determine the total vertex irregularity strength of comb product of two cycles and two stars.

#### Keywords

irregularity strength; comb product; total labeling

#### Full Text:

PDF

DOI: http://dx.doi.org/10.19184/ijc.2019.3.2.2

#### References

M. Baca, S. Jendrol, M. Miller, and J. Ryan, On irregular total labellings, Discrete Math. 307 (2007), 1378–1388.

P. Majerski and J. Przybylo, Total vertex irregularity strength of dense graphs, J. Graph Theory, 76(1) (2014), 34–41.

Nurdin, E.T.Baskoro, A.N.M.Salman and N.N.Goas, On the total vertex irregularity strength of trees, Discrete Math., 310 (2010), 3043–3048.

Nurdin, E.T. Baskoro, A.N.M. Salman and N.N. Goas, On the total vertex irregular labeling for several types of trees, Utilitas Mathematica., 83 (2010), 277–290.

Nurdin, A.N.M. Salman, N.N. Goas, and E.T. Baskoro, On the total vertex-irregular strength of a disjoint of t copies of a path, J. Combin. Math., 71 (2009), 227–233.

J.Przybylo, Liniear bound on the irregularity strength and the total vertex irregularity strength of graphs, SIAM J. Discrete Math. 23 (2009), 511–516

R. Ramdani, A.N.M. Salman, H. Assiyatun, A. Semaniˇ cová-Feˇ novciˇ cová, M. Baˇ ca, On the total irregularity strength of disjoint union of arbitrary graphs, Mathematical Reports, 18(68), 4 (2016), 469–482.

R. Ramdani and M.A. Ramdhani, Total vertex irregularity strength of comb product of two cycles, MATEC Web of Conference, 197, 01007 (2018).

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. Jendro ˇ l, Total vertex irregular labeling of complete bipartit graphs, J. Combin. Math. Combin. Comput., 55 (2005), 129–136.

### Refbacks

• There are currently no refbacks.

ISSN: 2541-2205