Total vertex irregularity strength of comb product of two cycles

Let 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 (G) ∪ E(G) → {1,2⋯,k}. The vertex weight v under the labeling f is denoted by wwff(v) and defined by wwff(v) = f(v) + ∑ ff(uuuu) uuuu∈EE(GG) . 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 labeling was introduced by Bača, 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, in the vertex oo, denoted by G⊳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 CCnn and CCmm where mm ∈ {1,2}.


Introduction
Mathematics is a logical, analytical, and systematic thinking framework in helping human to solve their problems.One of the topics in mathematics that is interesting to study is graph labeling.One of the applications of graph labeling are to solve network problem.
In this paper, we discuss a kind of graph labeling, which is total vertex irregular labeling.
Let G = (V,E) be a graph and k be a positive integer.A total k-labeling of G is a map f : V (G) ∪ E(G) → {1,2, ⋯,k}.
The notation   () is called by the weight of edge e under the labeling f.Definition 1.2.[1] The minimum  for which a graph  has an edge irregular total -labeling, denoted by (), is called the total edge irregularity strength of .
Bac ̌a et al [1] gave a lower bound and an upper bound on () for arbitrary graph G.The bounds is given by Theorem 2.1.
Theorem 1.1 [1] Let  = ( , ) be a graph with vertex set  and a non-empty edge set .Then In the same paper, they also gave the exact value of () for  are paths, cycles, stars, complete graphs, wheels, and friendships.[1] Ivanc ̌o and S. Jendrol' [2] posed a conjecture that for arbitrary graph  ≠  5 , Ramdani, et al [3] gave an upper bound on the total edge irregularity strength of disjoint union of graphs as follows.
Theorem 1.2 [3] The total edge irregularity strength of disjoint union of graphs  1 ,  2 , ⋯,   ,  ≥ 2, is In [4], Nurdin et al. determined the total edge irregular strength of the corona product of paths with some graphs.
Definition 1.3.[1] For an integer , a total labelling  :  ∪  → {1, 2, ⋯ } is called a vertex irregular total -labelling of  if every two distinct vertices  and  in  satisfy   () ≠   (), where   () = () + ∑ () ∈ .The notation   () is called by the weight of vertex u under the labeling f.Definition 1.4.[1] The minimum  for which a graph  has a vertex irregular total -labeling, denoted by (), is called the total vertex irregularity strength of .
In [1], Bac ̌a et al determined a lower bound and an upper bound on the total vertex irregularity strength of arbitrary graph  with  vertices,  edges, the minimum degree , and the maximum degree ∆, as follows.https://doi.org/10.1051/matecconf/201819701007AASEC 2018 In [5], Nurdin et al. determined another lower bound of () for G a connected graph as follows.Theorem 1.3 [5] Let  be a connected graph having   vertices of degree ( = ,  + 1,  + 2, ⋯ , ∆), where  and ∆ are the minimum and the maximum degree of  , respectively.Then Ramdani et al., in [6], determined an upper bound on the total vertex irregularity strength of Cartesian product of  2 and arbitrary regular graph .Theorem 1.4 [6].Let  be an -regular graph for  ≥ 1.Then In [7] and [8], Ramdani et al. determined an exact value of () for  are ladders and books.Nurdin et al. [9] gave the exact values of the total vertex irregularity strength for several types of trees and disjoint union of paths.Przybylo, in [10], gave a linear bound on ().
Combining edge and vertex irregular total labeling, Marzuki et al., in [11], introduced new irregular labeling, namely totally irregular total labeling, defined as follows.
Theorem 1.5 [12] Let  be a graph of order .Other results about () were given by Ramdani et al in [13].In the paper, they gave the exact value of () for  are Cartesian product of  2 and some families of graphs.In [14], Ramdani et al. gave the total irregularity strength of freindship.
In [15], Ramdani et al. determined the total irregularity strength of regular graphs.In the paper, they gave an upper bound on total irregularity strength of  copies of arbitrary graph .Also they gave the exact value of the total irregularity strength of  copies of path  2 .
In this paper, we determine the exact value of the total vertex irregularity strength of comb product of two cycles.The definition of comb product graph is given below.Definition 1.7 Let G and H be two connected graphs.Let o be a vertex of H .The comb product between G and H, in the vertex , denoted by G ⊳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.

Main Results
In this paper, we give the exact value of the total vertex irregularity strength of   ⊳o  4 and   ⊳o  5 .The first result is given by Theorem 2.1 Theorem 2.1 Let   be a cycle with n vertices and  4 be a cycle with 4 vertices, then for  ≥ 3, (  ⊳o  4 ) =  + 1. ( Proof.Let the vertex set of   ⊳o  4 be and the edge set be The ilustration of graph   ⊳o  4 can be seen in the Fig. 1. ( The labeling f gives weight of each vertex, for 1 ≤  ≤ , as follows : It can be seen that there are no two vertices of the same weight.So, f is a total vertex irregular labeling of   ⊳o  4 .The maximum label of the labeling f is  + 1.So, we have (  ⊳o  4 )≤  + 1. (29) From ( 18) and (29), we can conclude that for  ≥ 3, (  ⊳o  4 )=  + 1.
(30) ∎ This paper also determine the total vertex irregularity strength of   ⊳o  5 .The result can be seen in the theorem below.Theorem 2.2 Let   be a cycle with n vertices and  5 be a cycle with 5 vertices.For  ≥ 3, (  ⊳o  5 ) = ⌈ (53) It can be seen that there are no two vertices of   ⊳o  5 under labeling  of the same weight.
So that, f is a total vertex irregular labeling of   ⊳o 5.The maximum label of labeling f is ⌈ for  ≥ 3. ∎ Let the sequence   of Fibonacci numbers be defined by the recurrence relation   =  −1 +  −2 , n ≥ 3, with seed values  1 = 1 and  2 = 2. Then the total irregularity strength of a graph G is ts(G) ≤   .(8) In the same paper, [12], Ramdani et al. determined the exact value of () for G are gears, fungus, and  copies of stars.