Computing The Irregularity Strength of Planar Graphs

Hong Yang 1, Muhammad Kamran Siddiqui 2,* , Muhammad Ibrahim 3 and Sarfraz Ahmad 4 and Ali Ahmad 5 1 School of Information Science and Engineering, Chengdu University, Chengdu 610106, China; yanghong01@cdu.edu.cn 2 Department of Mathematics, COMSATS University Islamabad, Sahiwal Campus, Sahiwal 57000, Pakistan 3 Center for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University Multan, Multan 60800, Pakistan; mibtufail@gmail.com 4 Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan; sarfrazahmad@cuilahore.edu.pk 5 College of Computer Science & Information Systems, Jazan University, Jazan 45142, Saudi Arabia; ahmadsms@gmail.com * Correspondence: kamransiddiqui75@gmail.com


Introduction
All graphs considered here are finite, undirected, without loops or multiple edges.Denote by V(G) and E(G) the set of vertices and the set of edges of a graph G, respectively.Let |V(G)| = n and |E(G)| = m.
A labeling of a graph is any mapping that sends some set of graph elements to a set of numbers or colors.Graph labeling provides valuable information used in several application areas (see [1]).It is interesting to consider labeling the elements of the graph by the elements of a finite field.
For a graph G, we characterize a labeling ζ : V ∪ E → {1, 2, . . ., k} to be total k-labeling.A total k-labeling is characterized to be an edge irregular total k−labeling of the graph G if for each two distinct edges rs and r s their weights φ(r) + φ(rs) + φ(s) and φ(r ) + φ(r s ) + φ(s ) are distinct.In addition, total k-labeling is characterized to be a vertex irregular total k-labeling of the graph G if for each two distinctive vertices r and s their weights wt(r) and wt(s) are distinct.Here, the weight of a vertex r in G is the sum of the label of r and the labels of all edges incident with the vertex r.The least k for which the graph G has an edge irregular total k−labeling is called the total irregularity strength of G, represented by tes(G).Analogously, the minimum k for which the graph G has a vertex irregular total k−labeling is called the total vertex irregularity strength of G, denoted by tvs(G).
Chartrand et al. [2] introduced two graph invariants namely irregular assignments and the irregularity strength.Baca et al. [3] modified these graph invariants and introduced the concept of total edge irregularity strength and total vertex irregularity strength for a graph G.A simple lower bound for tes(G) and tvs(G) of a (p, q)-graph G in terms of maximum degree ∆(G) and the minimum degree δ(G), determine in the following theorems.Theorem 1. [3] Let G be a finite graph with p vertices, q edges and having maximum degree ∆ = ∆(G), the upper square brackets represent the ceiling function, and then Theorem 2. [3] Let G be a finite graph with p vertices, q edges, minimum degree δ = δ(G) and maximum degree ∆ = ∆(G), the upper square brackets represent the ceiling function, and then In [4], Ivančo and Jendrol' posed the following conjecture: Conjecture 1. [4] Let G be a finite graph with p vertices, q edges, different from K 5 with minimum degree δ = δ(G), maximum degree ∆ = ∆(G), the upper square brackets represent the ceiling function, and then In [5], Nurdin et al. posed the following conjecture: Conjecture 2. [5] Let G be a connected graph having n i vertices of degree i(i = δ, δ + 1, δ + 2, . . ., ∆), where δ and ∆ are the minimum and the maximum degree of G, respectively.Moreover, the upper square brackets represent the ceiling function, and then Conjecture 1 has been shown for complete graphs and complete bipartite graphs [6,7], for hexagonal grid graphs [8] , for toroidal grid [9], for generalized prism [10], for strong product of cycles and paths [11], for categorical product of two cycles [12], for zigzag graphs [13] and for strong product of two paths [14].
Conjecture 2 has been verified for for circulant graphs [15].Combining both total edge irregularity strength and total vertex irregularity strength notions, Marzuki et al. [16] introduced a new irregular total k-labeling of a graph G, which is required to be at the same time both vertex and edge irregular as follows: Definition 1.A total labeling φ : V ∪ E → {1, 2, . . ., k} is called totally irregular total k-labeling of G if every two distinct vertices u and v in V(G) satisfy wt(u) = wt(v), and every two distinct edges u 1 u 2 and v 1 v 2 in E(G) satisfy wt(u φ(uv) and wt(u 1 u 2 ) = φ(u 1 ) + φ(u 1 u 2 ) + φ(u 2 ).The minimum k for which a graph G has a totally irregular total k-labeling is called the total irregularity strength of G, denoted by ts(G).
Marzuki, et al. [16] gave a lower bond of ts(G) as follows: For every graph G, ts(G) ≥ max{tes(G), tvs(G)} (1) Ramdani and Salman [17] showed that the lower bound in Equation (1) for some cartesian product graphs is tight.Besides that, they determined the total irregularity strength of cycles and paths.For more details, see [18][19][20].In [21], Ahmad et al. found the exact value of total irregularity strength of generalized Petersen graph.
Example 1.For illustration, the concept of the totally irregular total k-labeling, we give an example from our recent paper [21] in which we show the totally irregular total 10-labeling for generalized Petersen graph P(9, 2) (see Figure 1).The weights for all vertices and the weights for all edges under the totally irregular total 10-labeling are given in Figure 2. Now, from Figure 2, it is easy to check that edge weights are different and represented by blue.On the other hand, the vertex weights are different and represented by black.
In this paper, we investigate the total irregularity strength of planar graphs.

The Planar Graph T n
Siddiqui introduced the planar graph S n in [22] and computed the tes(S n ), tvs(S n ).The planar graph T n (see Figure 3) is obtained from the planar graph S n by adding new edges x i y i+1 and having the same vertex set.The planar graph T n has Clearly, the planar graph T n has 3n vertices and 6n edges.More preciously, we call the cycle induced by {z i : 1 ≤ i ≤ n} the inner cycle, cycle induced by {y i : 1 ≤ i ≤ n} the outer cycle, and the set vertices {x i : 1 ≤ i ≤ n}, the outer vertices.All subscripts are taken under modulo n.In the next theorem, we determine the total irregularity strength of the planar graph T n .From Equation (1), we get ts(T n ) ≥ 6n+2

3
. For this, we define a and compute the vertex weight and edge weights in the following way.
Now, the weight of the edges and vertices of T n under the labeling φ are distinct.It is easy to check that there are no two edges of the same weight and there are no two vertices of the same weight.Thus, φ is a totally irregular total k−labeling.We conclude that ts(T n ) = 6n+2 3 , which complete the proof.

The Planar Graph R n (Pentagonal Circular Ladder)
In [23], Bača defined the prism D n (Circular ladder) for n ≥ 3. It is a cubic graph which can be defined as the cartesian product P 2 × C n on a path on two vertices with a cycle on n vertices.Prism D n , n ≥ 3 is considered of n−cycle y 1 , y 2 , y 3 , . . ., y n , an inner n−cycle x 1 , x 2 , x 3 , . . ., x n , and a set of n spokes y i z i , i = 1, 2, . . ., n, |V(D n | = 2n, |E(D n | = 3n.The planar graph (pentagonal circular ladder) R n (see Figure 4) is obtained from the graph of prism D n by adding a new vertex x i between y i and y i+1 , for i = 1, 2, 3, .., n.The planar graph (pentagonal circular ladder) R n has  For our purpose, we call the cycle induced by {z i : 1 ≤ i ≤ n} the inner cycle, and the cycle induced by {y i : 1 ≤ i ≤ n} ∪ {x i : 1 ≤ i ≤ n} the outer cycle.All subscripts are taken under modulo n.In the next theorem, we determine the total irregularity strength of the planar graph R n . .For this, we define a total labeling φ For n = 4, we have, For n = 5, we have, wt(x 2 ) = 4, wt(x 3 ) = 5, wt(x 4 ) = 6, wt(x 5 z) = 12, wt(y 1 ) = 13, wt(y 2 ) = 9, wt(y 3 ) = 10, wt(y 4 ) = 11, wt(y 5 ) = 15, wt(z 1 ) = 23, wt(z 2 ) = 19, wt(z 3 ) = 21, wt(z 4 ) = 24, wt(z 5 ) = 30, For n = 7, we have, For n = 10 , we have, For n ≥ 6 and n = 7, 10, we have φ( The weight of the edges and vertices under the labeling φ are distinct.It is easy to check that there are no two edges of the same weight and there are no two vertices of the same weight.Thus, φ is a totally irregular total k−labeling.We conclude that ts(R n ) = 4n+2 3 , which complete the proof.

The Planar Graph Q n
In [23], Bača defined the planar graph (pentagonal circular ladder) R n .The planar graph Q n (see Figure 5) is obtained from the planar graph (pentagonal circular ladder) R n by adding new edges y i y i+1 , z i w i , w i w i+1 .The planar graph Q n has The planar graph Q n has 4n vertices and 7n edges.For our purpose, we call the cycle induced by {w i : 1 ≤ i ≤ n} the inner cycle, the cycle induced by {z i : 1 ≤ i ≤ n} the middle cycle, the cycle induced by {y i : 1 ≤ i ≤ n} the outer cycle, and the set of vertices {x i : 1 ≤ i ≤ n} the set of outer vertices.The subscript n + 1 must be replaced by 1.For this, we define a total labeling φ from V(Q n ) ∪ E(Q n ) into 1, 2, . . ., 7n+2 3 and compute the vertex weight and edge weights in the following way.+ 34, for i = n The weight of the edges and vertices of S n under the labeling φ are distinct.It is easy to check that there are no two edges of the same weight and there are no two vertices of the same weight.Thus, φ is a totally irregular total k−labeling.We conclude that ts(Q n ) = 7n+2 3 , which completes the proof.

Conclusions
In this paper, we discus the total edge irregular k labeling, total vertex irregular k labeling and totally irregular total k labeling of planar graphs.We provide exact result of total irregularity strength ts for the planar graph T n , the planar graph R n (Pentagonal Circular Ladder) and the planar graph Q n .In the future, we are interested in computing the total irregularity strength ts for the other planar graphs.

3 and 3 and 1
compute the vertex weight and edge weights in the following way.Let k = 4n+2 ≤ i ≤ n.