Irregularity and Modular Irregularity Strength of Wheels

It is easily observed that the vertices of a simple graph cannot have pairwise distinct degrees. This means that no simple graph of the order of at least two is, in this way, irregular. However, a multigraph can be irregular. Chartrand et al., in 1988, posed the following problem: in a loopless multigraph, how can one determine the fewest parallel edges required to ensure that all vertices have distinct degrees? This problem is known as the graph labeling problem and, for its solution, Chartrand et al. introduced irregular assignments. The irregularity strength of a graph G is known as the maximal edge label used in an irregular assignment, minimized over all irregular assignments. Thus, the irregularity strength of a simple graph G is equal to the smallest maximum multiplicity of an edge of G in order to create an irregular multigraph from G. In the present paper, we show the existence of a required irregular labeling scheme that proves the exact value of the irregularity strength of wheels. Then, we modify this irregular mapping in six cases and obtain labelings that determine the exact value of the modular irregularity strength of wheels as a natural modification of the irregularity strength.


Introduction
Let G be a simple graph. Given a function ψ : E(G) → {1, 2, . . . , k}, the weight of a vertex x is wt ψ (x) = ∑ y∈N(x) ψ(xy), where N(x) denotes the set of neighbors of x in G. Such a function ψ we call an irregular assignment if wt ψ (x) = wt ψ (y) for all vertices x, y ∈ V(G) with x = y. The irregularity strength s(G) of a graph G is known as the maximal integer k, minimized over all irregular assignments, and is set to ∞ if no such function is possible. Clearly, s(G) < ∞ if and only if G contains no isolated edge and has, at most, one isolated vertex.
The notion of the irregularity strength was first introduced by Chartrand et al. in [1]. There is also given a lower bound of this graph invariant in the form s(G) ≥ max n i +i−1 where n i denotes the number of vertices of degree i and ∆ is the maximum degree of the graph G.
Faudree and Lehel in [2] studied the irregularity strength of regular graphs and showed that s(G) ≤ n 2 + 9 for d-regular graphs G of order n, d ≥ 2. They conjectured that there exists an absolute constant C such that s(G) ≤ n d + C. This upper bound was sequentially improved in [3][4][5] and recently in [6].
For several families of graphs, the exact value of the irregularity strength has been determined, namely for paths and complete graphs [1], for cycles and Turán graphs [7], for generalized Petersen graphs [8], for trees [9] and for circulant graphs [10]. The most complete recent survey of graph labelings is [11].
In [12], the authors introduced a modification of an irregular assignment known as a modular irregular assignment. A function ψ : E(G) → {1, 2, . . . , k} of a graph G of order n is called a modular irregular assignment if the weight function λ : V(G) → Z n defined by λ(x) = wt ψ (x) = ∑ y∈N(x) ψ(xy) is bijective and is called the modular weight of the vertex x, where Z n is the group of integers modulo n. The modular irregularity strength, ms(G), is defined as the minimum k for which G has a modular irregular assignment. If there is no such labeling for the graph G, then the value of ms(G) is defined as ∞.
Clearly, every modular irregular labeling of a graph with no component of the order of, at most, two is also its irregular assignment. This gives a lower bound of the modular irregularity strength, i.e., if G is a graph with no component of the order of, at most, two, then s(G) ≤ ms(G). ( We have already mentioned that if a graph G contains an isolated edge or at least two isolated vertices, then s(G) = ∞. The next theorem gives an infinity condition for the modular irregularity strength of a graph. Theorem 1 ([12]). If G is a graph of order n, n ≡ 2 (mod 4), then G has no modular irregular labeling, i.e., ms(G) = ∞.
The exact values of the modular irregularity strength for certain families of graphs, namely paths, cycles, stars, triangular graphs and gear graphs, are determined in [12]. A fan graph F n , n ≥ 2, is a graph obtained by joining all vertices of a path P n on n vertices to a further vertex, called the center. For fan graphs F n of order n + 1, n ≥ 2, in [13], it is proven that , otherwise.
A wheel W n , n ≥ 3, is a graph obtained by joining all vertices of a cycle C n to a further vertex c, called the center. Thus, W n contains n + 1 vertices, say, c, x 1 , x 2 , . . . , x n , and 2n edges cx i , 1 ≤ i ≤ n, x i x i+1 , 1 ≤ i ≤ n − 1, and x n x 1 .
In the present paper, we determine the exact value of the irregularity strength and the exact value of the modular irregularity strength of wheels. The paper is organized as follows. First, we investigate the existence of an irregular assignment of wheels. We improve the main idea of the construction of an irregular assignment for fan graphs used in [13] and we construct an edge labeling with the desired irregular properties. The existence of such labeling proves the exact value of the irregularity strength of wheels. Next, we modify this irregular mapping of wheels in six cases, and, for each case, we determine the exact value of the modular irregularity strength.

Irregular Assignment of Wheels
In this section, we discuss the irregularity strength for wheels. According to the lower bound given in (1), we have that s(W n ) ≥ n+2 3 . To show that n+2 3 is also an upper bound of the irregularity strength of wheels, we construct an edge labeling and show that this labeling meets the required properties. Now, for n ≥ 5, we define the edge labeling ψ as follows:
Proof. Consider the edge labeling ψ of wheels W n , n ≥ 5, defined above. One can easily check that, for odd n, For even n, we verify that Observe that, under the edge labeling ψ, the edge labels of a wheel W n , n ≥ 5, are, at most, n+2 3 . This implies that the edge labeling ψ is an n+2 3 -labeling.
The next lemma provides values of weights of the vertices of wheels produced by the edge labeling ψ.
The next theorem gives the exact value of the irregularity strength of wheels.
Theorem 2. Let W n , n ≥ 3, be a wheel on n + 1 vertices. Then, Proof. Chartrand et al. in [1] proved that s(K n ) = 3 for each n ≥ 3. Since W 3 = K 4 , it follows that s(W 3 ) = 3. Figure 1 depicts an irregular 3-labeling of the wheel W 3 . According to the lower bound given in (1), we have that s(W n ) ≥ n+2 3 . Hence, s(W 4 ) ≥ 2. Suppose that there exists an irregular 2-labeling of W 4 . As the vertices x i , i = 1, 2, 3, 4, are of degree 3, then, under any 2-labeling, the smallest weight of a vertex, say x 1 , is at least 3 (this is realizable as the sum of edge labels 1 + 1 +1) and the largest weight of the vertex, which must be x 3 , is, at most, 6 (this is realizable as the sum of edge labels 2 + 2 +2). Since all vertices x i must have distinct weights, it follows that the weight of the center vertex is 6 and we have a contradiction. It follows that there is no irregular 2-labeling of W 4 . Figure 2 illustrates an irregular 3-labeling of the wheel W 4 .  > n(n+5) 6 > n + 2 for any n ≥ 5, we find that the center vertex weight wt ψ (c) > n + 2. It follows that the vertex weights are distinct for all pairs of distinct vertices and the labeling ψ is a suitable edge-irregular n+2 3 -labeling. This concludes the proof.

Modular Irregular Assignment of Wheels
For determining the exact value of the modular irregularity strength of wheels, we use a modular irregular assignment as a suitable modification of the irregular n+2 3 -labeling ψ. From Lemma 2, it follows that the weights of vertices x i , i = 1, 2, . . . , n, n ≥ 5, under the labeling ψ, gradually reach the values 3, 4, . . . , n, n + 1, n + 2, i.e., the modular weights are 3, 4, . . . , n, 0, 1 (mod n + 1). In order to obtain a modular irregular assignment of wheel W n , the center vertex modular weight has to be congruent to 2 (mod n + 1). To produce such a modular weight of the center vertex, we increase (decrease) the labels of edges of W n in such a way that these operations will have no impact on the weights of the vertices x i , i = 1, 2, . . . , n.
The following lemma gives the exact value of the modular irregularity strength of wheels W n , for even n ≥ 4.

Lemma 3.
Let W n be a wheel on n + 1 vertices with n ≥ 4 even. Then,

otherwise.
Proof. The existence of a modular irregular 3-labeling of the wheel W 4 follows from Figure 2, where the modular weights of vertices x i , i = 1, 2, 3, 4, are 1, 2, 4, 3 (mod 5) and the modular weight of the center vertex is 0 (mod 5). Next, for even n > 4, we will consider the following three cases. Case 1. n ≡ 0 (mod 6).
Define an edge labeling f 0 as modification of the labeling ψ in the following way.
It is a matter of routine checking that, for n = 6, the edge label f 0 (x 5 x 6 ) = ψ(x 5 x 6 ) + 1 = 3 = n+2 3 , and for n > 6, we have Moreover, decreasing the labels of the edges cx n , cx n−1 by one and the labels of the edges cx n−i , i = 2, 3, 4, . . . , n 3 − 2, n 3 − 1, by two, and increasing the label of the edge x n−1 x n by one and the labels of the edges x n−i x n−i+1 , i = 3, 5, 7, . . . , n 3 − 3, n 3 − 1, by two has no effect on the weights of vertices x i , 1 ≤ i ≤ n. They still successively attain values from 3 to n + 2. On the other hand, decreasing the labels of the edges cx n , cx n−1 and cx n−i , i = 2, 3, 4, . . . , n 3 − 2, n 3 − 1, decreases the center vertex weight and we obtain Since n 6 is an integer, it follows that wt f 0 (c) ≡ 2 (mod n + 1). This implies that the modified labeling f 0 is a required modular irregular n+2 3 -labeling of W n . Case 2. n ≡ 2 (mod 6). In light of Lemma 2, we can see that under the labeling ψ, the weight of the center vertex is and it is not congruent to 2 (mod n + 1). Therefore, we modify the edge labeling (3) as follows: The new labeling f 2 reduces the value of the edge x 2 x 3 by one and increases the values of edges cx 2 and cx 3 by one. This modification of the labeling ψ has no impact on the weights of the vertices x 2 and x 3 . However, increasing the values of the edges cx 2 and cx 3 results in an increase in the weight of the center vertex and we obtain wt f 2 (c) = wt ψ (c) + 2 = (n+1)(n+4) As n+4 6 is an integer, subsequently, wt f 2 (c) ≡ 2 (mod n + 1). This proves that the constructed edge labeling f 2 is a suitable modular irregular n+2 3 -labeling of W n . Case 3. n ≡ 4 (mod 6).
Let us first assume that wt ϕ (c) = 24. The sum of all the vertex weights of W 10 is In the computation of the vertex weights of W 10 , each edge label is used twice. Then, the sum of all edge labels used to calculate the vertex weights is equal to the sum of all the vertex weights. With respect to (4), we obtain Using parity considerations for the left-hand and the right-hand sides of Equation (5), we obtain a contradiction. Thus, there is no modular irregular 4-labeling of W 10 with wt ϕ (c) = 24. Now, let us assume that wt ϕ (c) = 35. Under a modular irregular 4-labeling ϕ of W 10 , if wt ϕ (x i ) = 3, then ϕ(cx i ) = 1. If wt ϕ (x j ) = 4, then ϕ(cx j ) is, at most, 2 and if wt ϕ (x k ) = 5, then ϕ(cx k ) is, at most, 3. Values of the other seven edges incident with the center can be, at most, 4. Thus the weight of the center cannot be more than 34 and, again, we have a contradiction. This proves that there is no modular irregular 4-labeling of W 10 . A modular irregular 5-labeling of W 10 is displayed in Figure 3.
For the wheel W 16 , we have a special modular irregular 6-labeling, given in Figure 4.
To obtain a required edge labeling with the property that the weights of vertices x i ∈ V(W n ) will not change and the weight of the center vertex will be congruent to two (mod n + 1), we construct an edge labeling f 4 of W n , n ≥ 22, such that for i = 2, n, 4, for i = 3, 6, for i = 4, It is not difficult to verify that increasing the labels of the edges cx 2 , cx n+2 3 +2 , cx n+2 3 +3 by one, the labels of the edges cx 3 , cx i , 5 ≤ i ≤ n+2 3 + 1, by two and the label of the edge cx 4 by three and decreasing the labels of the edges x 2 x 3 , x 3 x 4 , x n+2 3 +2 x n+2 3 +3 by one and the labels of the edges x i x i+1 , 4 ≤ i ≤ n+2 3 for even i has no impact on the weights of the vertices x i ∈ V(W n ), as they successively assume values from 3 to n + 2. However, increasing the labels of the edges cx i , 2 ≤ i ≤ n+2 3 + 3, results in an increase in the value of the center vertex and we obtain Because n+8 6 is an integer, wt f 4 (c) ≡ 2 (mod n + 1). Moreover, it is easy to check that In light of the previous discussion, it follows that for n ≥ 22, the labeling f 4 is the desired modular irregular n+2 3 -labeling of W n . x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 c  x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 13 x 14 x 15 x 16 c Figure 4. A modular irregular 6-labeling of W 16 .
Directly from Theorem 1, we achieve the following result. Corollary 1. If n ≡ 1 (mod 4), then the wheel W n on n + 1 vertices has no modular irregular labeling.
In the next lemma, we give the exact value of the modular irregularity strength of wheels W n , for odd n ≥ 3.

Lemma 4.
Let W n be a wheel on n + 1 vertices with odd n ≥ 3. Then, if n ≡ 1 (mod 4),  x 1 x 2 x 3 x 4 x 5 x 6 x 7 Figure 5. A modular irregular 3-labeling of W 7 .
It is a routine procedure to verify that the edge labeling f 7 does not increase the largest values of the edges in W n , has no effect on the weights of vertices x i ∈ V(W n ), but increases the weight of the center vertex by n+5 6 + 2. Thus, we obtain wt f 7 (c) = wt ψ (c) + n+5 6 + 2 = (n+1)(n+5) 6 + 2.