Non-inclusive and inclusive distance irregularity strength for the join product of graphs

A function φ : V ( G ) → { 1 , 2 , . . . , k } of a simple graph G is said to be a non-inclusive distance vertex irregular k -labeling of G if the sums of labels of vertices in the open neighborhood of every vertex are distinct and is said to be an inclusive distance vertex irregular k -labeling of G if the sums of labels of vertices in the closed neighborhood of each vertex are different. The minimum k for which G has a non-inclusive (resp. an inclusive) distance vertex irregular k -labeling is called a non-inclusive (resp. an inclusive) distance irregularity strength and is denoted by dis( G ) (resp. by (cid:99) dis( G ) ). In this paper, the non-inclusive and inclusive distance irregularity strength for the join product graphs are investigated.


Introduction
All graphs considered here are assumed to be simple, finite and undirected. Let G be a graph with vertex-set V (G) = V and edge-set E(G) = E. For a vertex v ∈ V , the degree of v, denoted by deg G (v), is the number of vertices adjacent to v. The open and closed neighborhood of v is defined as N G (v) = {u : uv ∈ E} and N G [v] = {v} ∪ N G (v), respectively. The maximum degree of vertices in G is denoted by ∆(G). By graph labeling we mean any mapping that carries some sets of graph elements to a set of non-negative integers, called labels. There are many types of graph labelings that have been developed. A survey of recent results on graph labelings is provided by Gallian [8].
Let k be a positive integer and let a graph G be given. A function φ : V → {1, 2, . . . , k} is said to be a non-inclusive distance vertex irregular k-labeling of G if the weights are distinct for every pair of two distinct vertices, where the weight of a vertex v is defined as the sum of labels of vertices in the open neighborhood of v in G. The non-inclusive distance irregularity strength of G, denoted by dis(G), is the minimum integer k for which G has a non-inclusive distance vertex irregular klabeling. Furthermore, the labeling φ is called an inclusive distance vertex irregular k-labeling of G if for each two vertices u and v, there is wt The least integer k for which G has an inclusive distance vertex irregular k-labeling is called the inclusive distance irregularity strength, dis(G). We will say that dis(G) = ∞ and dis(G) = ∞ whenever such a non-inclusive and an inclusive distance vertex irregular labeling does not exist, respectively.
The notion of non-inclusive distance vertex irregular labelings was intoduced in 2017 by Slamin [13]. Meanwhile, Bača et al. [3] developed inclusive distance vertex irregular labelings one year later as a variation of the non-inclusive irregularity strength of graphs. These graph invariants are then generalized by Bong et al. [5] to non-inclusive and inclusive d-distance irregularity strength of graphs where d is an integer arbitrarily taken from 1 up to diameter of the graph. Thus, a non-inclusive 1-distance vertex irregular labeling is called a non-inclusive distance vertex irregular labeling. Similarly, we call an inclusive 1-distance vertex irregular labeling as an inclusive distance vertex irregular labeling.
In the literature, it was investigated the total version of this concept, see [19,20]. Furthermore, related topics on the subjects can also be found in, for example, [1,6,9], and for some new results, see [2,10,12].
The following lemmas give the necessary and sufficient condition for a graph G to have finite dis(G) and dis(G).
In the present paper, we deal with a so-called product of graphs namely a join product. The join product of two graphs G and H, denoted by G ⊕ H, is a graph obtained from G and H by joining an edge from each vertex of G to each vertex of H. We represent the vertex-set of We here consider the following problems. Thus, in the rest of the paper, we will only deal with the case when dis(G ⊕ H) < ∞ and dis(G ⊕ H) < ∞.
We need to define some notations related to the non-inclusive distance irregularity strength of graphs as follows. Let G and H be graphs with dis(G) < ∞ and dis(H) < ∞. Let φ G and φ H be a non-inclusive distance vertex irregular dis(G)-labeling of G and a non-inclusive distance vertex irregular dis(H)-labeling of H, respectively. For a vertex v ∈ V (G) and a non-negative integer α, we define an α-weight of v under a labeling φ G of a graph G as We denote by v α max a vertex of G in such away that wt α respectively. Further, we also consider positive integers β G and γ G,H such that β G = max 1, max and respectively. With respect to the inclusive distance irregularity strength, we shall also define some notations as follows. Given two graphs G and H with dis(G) < ∞ and dis(H) < ∞, let φ G and φ H be an inclusive distance vertex irregular dis(G)-labeling of G and an inclusive distance vertex irregular dis(H)-labeling of H, respectively. Let α be a non-negative integer. We define an α-weight of a vertex v of G under a labeling φ G of a graph G as In particular, when α = 0, we will use, respectively, . Moreover, we also define positive integers β G and γ G,H such that and respectively. Let x and y be two given integers. Then we define

dis(G ⊕ H) and dis(G ⊕ H)
In this section, we give the construction of the non-inclusive and inclusive distance vertex irregular labeling for the join product graphs. Our basic idea is to construct a new non-inclusive distance vertex irregular labeling for the join product graphs G ⊕ H from the described noninclusive distance vertex irregular labeling of G and H. Similar ideas are then used to construct the inclusive distance vertex irregular labeling of the join product graphs G ⊕ H.
Our first result below provides the lower bound of the non-inclusive distance irregularity strength for the join product of two graphs in terms of dis(G) and dis(H).
Proof. We first show that there is no non-inclusive distance vertex irregular k-labeling of a graph G ⊕ H such that k < dis(G). Suppose to the contrary that such labeling φ exists, that is, a labeling φ : Since each vertex of G is adjacent to all the vertices of H and since all the vertices of G have distinct weights then if we subtract from all these weights the sum of labels of all vertices of H, it gives us a restriction of the labeling φ on the graph G which is a non-inclusive distance vertex irregular k -labeling of G for some k ≤ k. But this gives a contradiction as k ≤ k < dis(G).
Next we prove that there is no non-inclusive distance vertex irregular k-labeling φ of a graph G ⊕ H such that k < dis(H). Using similar arguments with the previous case we can obtain a restriction of the labeling φ on the graph H which is a non-inclusive distance vertex irregular k -labeling of H with k ≤ k, giving a contradiction as k ≤ k < dis(H).
The following lemma gives the sufficient condition for α-weights of all vertices in a graph to be different. Lemma 2.2. Let G be a graph with dis(G) < ∞ and let φ be a non-inclusive distance vertex irregular dis(G)-labeling of G. Let β G be an integer defined in (1). Then for any integer α ≥ β G and every two distinct vertices Proof. For some α and some However, on the other hand, as α ≥ β G , we have which gives us a contradiction. This proves the first part of the statement. Next we prove the second part of the statement. Here we use the similar technique as the first part. Thus we suppose to the contrary that for some α and some u , v ∈ V (G), However, on the other hand, as α ≥ β G , we obtain again a contradiction.
Notice that the property in Lemma 2.2 implies that for any integer Next, as γ G,H ≥ β G , the following property is satisfied according to Lemma 2.2. Corollary 2.1. Let G and H be graphs such that dis(G ⊕ H) < ∞, and let φ G and φ H be a non-inclusive distance vertex irregular dis(G)-labeling of G and a non-inclusive distance vertex irregular dis(H)-labeling of H, respectively. Let β G and γ G,H be integers defined in (1) and (2), respectively. Then for any two distinct vertices u, v ∈ V (G), wt The value of the non-inclusive distance irregularity strength for G⊕H is given in the following theorem.
Theorem 2.1. Let G and H be graphs such that dis(G ⊕ H) < ∞, and let φ G and φ H be a non-inclusive distance vertex irregular dis(G)-labeling of G and a non-inclusive distance vertex irregular dis(H)-labeling of H, respectively. If either Otherwise, Proof. We distinguish our proof into two cases.
Put k = max{dis(G), dis(H)}. Due to Lemma 2.1 it is enough to show that there exists a non-inclusive distance vertex irregular k-labeling of G ⊕ H. Let ϕ be a labeling on the vertices of G ⊕ H defined as follows.
Obviously the largest label appearing on the vertices under the labeling ϕ is k and the weights of the vertices are given by We show that the vertex weights are distinct for every two vertices u, v ∈ V (G ⊕ H). If both u and v are in We now suppose that u ∈ V (G) and v ∈ V (H). The condition (i) implies that wt ϕ (v max ) < wt ϕ (u min ) which means that wt ϕ (u) = wt ϕ (v). Similarly, the restriction (ii) implies that wt ϕ (u max ) < wt ϕ (v min ) meaning that wt ϕ (u) = wt ϕ (v).
Clearly the labels used on the labeling ϕ 1 are at most k 1 . For the vertex weights we have We show that for every two distinct vertices u and v of G ⊕ H, wt . We now consider u ∈ V (G) and v ∈ V (H). It suffices for us to show that wt

Using these informations together with the facts that
and y x y + 1 > x, we get or equivalently wt ϕ 1 (u min ) > wt ϕ 1 (v max ). Thus ϕ 1 is a non-inclusive distance vertex irregular k 1 -labeling of G ⊕ H and hence dis(G ⊕ H) ≤ k 1 . Analogously, we define another vertex k 2 -labeling ϕ 2 of G ⊕ H as follows.
Using similar arguments with the previous one we can obtain that ϕ 2 is a non-inclusive distance vertex irregular k 2 -labeling of G ⊕ H and hence dis(G ⊕ H) ≤ k 2 . Taking the minimum from both k 1 and k 2 , it brings us to the desired result.
The following results related to the inclusive distance irregularity strength are presented. The proofs are omitted since ideas similar with Lemmas 2.1 and 2.2, Corollary 2.1 and Theorem 2.1, respectively, are used as arguments.
Corollary 2.2. Let G and H be graphs such that dis(G ⊕ H) < ∞, and let φ G and φ H be an inclusive distance vertex irregular dis(G)-labeling of G and an inclusive distance vertex irregular dis(H)-labeling of H, respectively. Let β G and γ G,H be integers defined in (3) and (4), respectively. Then for any two distinct vertices u, v ∈ V (G), wt Theorem 2.2. Let G and H be graphs such that dis(G ⊕ H) < ∞, and let φ G and φ H be an inclusive distance vertex irregular dis(G)-labeling of G and an inclusive distance vertex irregular dis(H)-labeling of H, respectively. If either Otherwise, dis(G ⊕ H) ≤ min max dis(G), dis(H) + γ H,G , max dis(H), dis(G) + γ G,H .
If we take H ∼ = K 1 then from Theorem 2.2 we obtain the inclusive distance irregularity strength for the graph G ⊕ K 1 which was proved by Bača et al. [3].

dis(G ⊕ K 1 )
In [4], Bong et al. showed that the non-inclusive distance irregularity strength of G ⊕ K 1 and G is equal as stated in the following theorem.
However, the above assertion is not true as we can easily see a counter example namely the complete graph K n ∼ = K n−1 ⊕ K 1 of Slamin [13] which showed that dis(K n ) = dis(K n−1 ⊕ K 1 ) = n = n − 1 = dis(K n−1 ).
In this section, we provide a correction for Theorem 3.1. We prove that dis(G ⊕ K 1 ) can be either dis(G) or dis(G) + 1. We will need the following lemma in order to prove our theorem.
Next we show that m = dis(G). By Lemma 2.1, m ≤ dis(G). Now assume that m < dis(G). Then a labeling φ G on the vertices of G defined as where p ∈ {1, 2, . . . , dis(G)}\{φ G (u) : u ∈ V (G)\V (G * )}, is a non-inclusive distance vertex irregular dis(G)-labeling of G. Next let u max ∈ V (G) (possibly u max = u max ) such that yielding a contradiction. Hence m = dis(G).
Now we are ready to prove the main result of this section. Note that for each graph G with dis(G) < ∞ and non-inclusive distance vertex irregular labeling φ G , it holds that Theorem 3.2. Let G be a graph with dis(G) < ∞. If there exists a non-inclusive distance vertex irregular dis(G)-labeling φ G of G such that u∈V (G) φ G (u) > wt φ G (u max )+1 then dis(G⊕K 1 ) = dis(G). Otherwise dis(G ⊕ K 1 ) = dis(G) + 1.
Proof. The first case follows from Theorem 2.1. Now we consider the second case, i.e., for every non-inclusive distance vertex irregular dis(G)- On the other hand, the labeling ϕ defined below is a non-inclusive distance vertex irregular (dis(G) + 1)labeling of H,
We begin with the following observation which is easy to prove. The next lemma presents the upper bound for the inclusive distance irregularity strength of complete multipartite graphs with same size of partite sets. where n, p ≥ 2. Then dis(G) ≤ n + 2(p − 1).
The following result supports Conjecture 1.
Proof. The upper bound follows from Theorem 4.1 and the lower bound is obtained from Lemma 2.3.