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

#### Abstract

*G*= (

*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

*G*\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

#### Full Text:

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

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