Modular H-Irregularity Strength of Graphs

Abstract

Two new graph characteristics, the modular edge H-irregularity strength and the modular vertex H-irregularity strength, are introduced. Lower bounds on these graph characteristics are estimated, and their exact values are determined for certain families of graphs. This demonstrates the sharpness of the presented lower bounds.

1. Introduction

Consider a simple graph $G=(V,E)$ with vertex set $V\left(G\right)$ and edge set $E\left(G\right)$. A labeling of a graph is a mapping that assigns numbers (usually positive integers) to its elements. The most common domains are the vertex set (yielding vertex labelings) and the edge set (yielding edge labelings).

An edge-covering of G is a family of subgraphs $H_1,H_2,\dots ,H_t$ such that each edge of $E\left(G\right)$ belongs to at least one of the subgraphs $H_i$, $i=1,2,\dots ,t$. In this case, we say that G admits an $(H_1,H_2,\dots ,H_t)$-(edge) covering. If each subgraph $H_i$ is isomorphic to a given graph H, then the graph G is said to admit an H-covering.

Let G be a graph admitting an H-covering. For a subgraph $H\subseteq G$, under an edge k-labeling $\phi :E\left(G\right)\to \{1,2,\dots ,k\}$, where k is a positive integer, the associated H-weight is defined by

$$w{t}_{\phi }\left(H\right)=\sum _{e\in E\left(H\right)}\phi \left(e\right).$$

An edge k-labeling $\phi$ is called an H-irregular edge k-labeling of the graph G if for every pair of distinct subgraphs $H_1$, $H_2$ isomorphic to H, it holds that $w{t}_{\phi }\left(H_1\right)\ne w{t}_{\phi }\left(H_2\right)$. The smallest integer k for which such a labeling exists is called the edge H-irregularity strength of G and is denoted by $\mathrm{ehs}(G,H)$.

Similarly, under a vertex k-labeling $\psi :V\left(G\right)\to \{1,2,\dots ,k\}$, the associated H-weight is defined by

$$w{t}_{\psi }\left(H\right)=\sum _{v\in V\left(H\right)}\psi \left(v\right).$$

A vertex k-labeling $\psi$ is called an H-irregular vertex k-labeling of G if for every pair of distinct subgraphs $H_1$, $H_2$ isomorphic to H, there is $w{t}_{\psi }\left(H_1\right)\ne w{t}_{\psi }\left(H_2\right)$. The vertex H-irregularity strength of G, denoted by $\mathrm{vhs}(G,H)$, is the minimum of the maximum label k over all such labelings. If no such labeling exists, we write $\mathrm{vhs}(G,H)=\infty$.

The concepts of the edge H-irregularity strength and the vertex H-irregularity strength were introduced by Ashraf et al. in [1], who provided lower bounds for the parameter $\mathrm{ehs}(G,H)$ and $\mathrm{vhs}(G,H)$ as follows:

Theorem 1

([1]). Let G be a graph admitting an H-covering given by t subgraphs isomorphic to H. Then,

$$\begin{array}{c}\hfill \mathrm{ehs}(G,H)\ge ⌈1+\frac{t-1}{\left|E\right(H\left)\right|}⌉,\end{array}$$

(1)

$$\begin{array}{c}\hfill \mathrm{vhs}(G,H)\ge ⌈1+\frac{t-1}{\left|V\right(H\left)\right|}⌉.\end{array}$$

(2)

The exact values of $\mathrm{ehs}(G,H)$ and $\mathrm{vhs}(G,H)$ for paths, fans and ladders were determined in [1,2], establishing the sharpness of bounds (1) and (2).

Theorem 2

([1]). Let n and m be positive integers, $2\le m\le n$. Then,

$$\mathrm{ehs}(P_n,P_m)=⌈{\displaystyle \frac{n-1}{m-1}}⌉and\mathrm{vhs}(P_n,P_m)=⌈{\displaystyle \frac{n}{m}}⌉.$$

Theorem 3

([2]). Let $F_n$, $n\ge 2$, be a fan graph on $n+1$ vertices admitting $F_m$-covering, where m is a positive integer, $2\le m\le n$. Then,

$$\mathrm{ehs}(F_n,F_m)=⌈\frac{n+m-1}{2m-1}⌉and\mathrm{vhs}(F_n,F_m)=⌈\frac{n+1}{m+1}⌉.$$

Theorem 4

([2]). Let $L_n\cong P_n\square P_2$, $n\ge 2$, be a ladder admitting $L_m$-covering, where m is a positive integer, $2\le m\le n$. Then,

$$\mathrm{ehs}(L_n,L_m)=⌈\frac{2m+n-2}{3m-2}⌉and\mathrm{vhs}(L_n,L_m)=⌈\frac{m+n}{2m}⌉.$$

Several papers have been published in this area over the past four years. For example, the exact values of the edge H-irregularity strength of hexagonal and octagonal grid graphs were computed in [3], and in [4], the vertex (edge) H-irregular labelings of graphs G are investigated, where both G and H are either a comb product or an edge comb product of graphs.

If H is isomorphic to $K_2$ then a $K_2$-irregular vertex k-labeling is equivalent to an edge irregular k-labeling. Hence, the vertex $K_2$-irregularity strength $\mathrm{vhs}(G,K_2)$ is the edge irregularity strength $\mathrm{es}\left(G\right)$. This graph characteristic was introduced by Ahmad et al. [5], who provided lower bounds and exact values for paths, stars and the Cartesian product of two paths. Additional results on the edge irregularity strength can be found in [6,7,8]. The edge irregularity strength is an edge modification of the concept of the irregularity strength $s\left(G\right)$, originally introduced by Chartrand et al. in [9] and further studied in [10,11,12,13,14].

A natural variation of the H-irregular edge (vertex) k-labeling is a modular H-irregular edge (vertex) k-labeling. Suppose that G admits an $(H_1,H_2,\dots ,H_t)$-covering, where each subgraph $H_i$ is isomorphic to a given graph $H\subseteq G$. An edge labeling $\phi :E\left(G\right)\to \{1,2,\dots ,k\}$ is called a modular H-irregular edge k-labeling if the weight function $\lambda :\{H_1,H_2,\dots ,H_t\}\to {\mathbb{Z}}_t$ given by $\lambda \left(H_i\right)=w{t}_{\phi }\left(H_i\right)={\displaystyle \sum _{e\in E\left(H_i\right)}}\phi \left(e\right)$ is bijective, where ${\mathbb{Z}}_t$ is the group of integers modulo t. The value $w{t}_{\phi }\left(H_i\right)$ is called the modular weight of the subgraph $H_i$. The modular edge H-irregularity strength, $\mathrm{mehs}(G,H)$, is defined as the smallest integer k for which such a labeling exists.

Analogously, a vertex labeling $\psi :V\left(G\right)\to \{1,2,\dots ,k\}$ is called a modular H-irregular vertex k-labeling if the weight function $\lambda :\{H_1,H_2,\dots ,H_t\}\to {\mathbb{Z}}_t$ defined by $\lambda \left(H_i\right)=w{t}_{\psi }\left(H_i\right)={\displaystyle \sum _{v\in V\left(H_i\right)}}\psi \left(v\right)$ is bijective. Again, $w{t}_{\psi }\left(H_i\right)$ is called the modular weight of the subgraph $H_i$. The modular vertex H-irregularity strength, $\mathrm{mvhs}(G,H)$, is defined as the minimum of the maximum vertex label k over all such labelings.

Note that the modular $K_2$-irregular vertex k-labeling of a graph G is equivalent to a modular edge irregular k-labeling of G. Therefore, the modular vertex $K_2$-irregularity strength $\mathrm{mvhs}(G,K_2)$ is the modular edge irregularity strength $\mathrm{mes}\left(G\right)$ introduced in [15], which also provides exact values for caterpillars, cycles, and friendship graphs. The modular irregularity strength $\mathrm{ms}\left(G\right)$ as an edge modification of the modular edge irregularity strength $\mathrm{mes}\left(G\right)$ was introduced in [16], where precise values of the modular irregularity strength for paths, cycles and stars are determined. The exact values of the modular irregularity strength for fan graphs and wheels are addressed in [17] and [18], respectively.

The main objective of the present paper is to estimate lower bounds on the modular edge H-irregularity strength and the modular vertex H-irregularity strength, to investigate the existence of corresponding labelings for various families of graphs, and to determine exact values of these parameters, thereby demonstrating the sharpness of the proposed bounds.

2. Lower Bounds for New Graph Characteristics

Evidently, every modular H-irregular edge k-labeling of a graph G is also an H-irregular edge k-labeling of G, and similarly, every modular H-irregular vertex k-labeling of G is also an H-irregular vertex k-labeling of G. Accordingly, for any simple graph G, the following inequalities hold:

$$\begin{array}{c}\hfill \mathrm{ehs}(G,H)\le \mathrm{mehs}(G,H),\end{array}$$

(3)

$$\begin{array}{c}\hfill \mathrm{vhs}(G,H)\le \mathrm{mvhs}(G,H).\end{array}$$

(4)

Of course, in general, the converses of (3) and (4) do not hold. Nevertheless, the validity of the following claims is immediate.

Theorem 5.

Let G be a simple graph admitting an H-covering with $\mathrm{ehs}(G,H)=k$. If H-weights under the corresponding H-irregular edge k-labeling form a sequence of consecutive integers, then

$$\begin{array}{c}\hfill \mathrm{ehs}(G,H)=\mathrm{mehs}(G,H)=k.\end{array}$$

(5)

Theorem 6.

Let G be a simple graph admitting an H-covering with $\mathrm{vhs}(G,H)=k$. If H-weights under the corresponding H-irregular vertex k-labeling form a sequence of consecutive integers, then

$$\begin{array}{c}\hfill \mathrm{vhs}(G,H)=\mathrm{mvhs}(G,H)=k.\end{array}$$

(6)

The existence of a $P_m$-irregular edge $⌈\frac{n-1}{m-1}⌉$-labeling and a $P_m$-irregular vertex $⌈\frac{n}{m}⌉$-labeling of the path on n vertices is established in the proof of Theorem 2, where the corresponding $P_m$-weights form a sequence of consecutive integers in both cases. In light of Theorems 5 and 6, we obtain the following corollary.

Corollary 1.

Let $P_n$, $n\ge 2$, be a path on n vertices and let m be a positive integer such that $2\le m\le n$. Then,

$$\mathrm{mehs}(P_n,P_m)=⌈\frac{n-1}{m-1}⌉and\mathrm{mvhs}(P_n,P_m)=⌈\frac{n}{m}⌉.$$

The fan graph $F_n$, where $n\ge 2$, is the graph obtained by joining each vertex of a path on n vertices to an additional vertex. In the proof of Theorem 3, the existence of an $F_m$-irregular edge $⌈\frac{n+m-1}{2m-1}⌉$-labeling of $F_n$ is established, where the corresponding $F_m$-weights form a sequence of consecutive integers. By Theorem 5, we obtain the following corollary:

Corollary 2.

Let $F_n$, $n\ge 2$, be a fan graph on $n+1$ vertices and let m be a positive integer such that $2\le m\le n$. Then,

$$\mathrm{mehs}(F_n,F_m)=⌈\frac{n+m-1}{2m-1}⌉.$$

To obtain the modular vertex $F_m$-irregularity strength of a fan graph, we first establish the following general theorem concerning the join of two graphs. The join $G+F$ of two disjoint graphs G and F is defined as the union of G and F, together with all edges connecting each vertex of G to each vertex of F.

Theorem 7.

Let G and F be disjoint graphs such that G admits an H-covering consisting of exactly t subgraphs isomorphic to H. If the graph $G+F$ contains exactly t subgraphs isomorphic to $H+F$, then

$$\mathrm{vhs}(G,H)=\mathrm{vhs}(G+F,H+F).$$

Moreover, if there exists an $(H+F)$-irregular vertex $\mathrm{vhs}(G,H)$-labeling of $G+F$, where the corresponding $(H+F)$-weights form a sequence of consecutive integers, then

$$\mathrm{mvhs}(G,H)=\mathrm{mvhs}(G+F,H+F).$$

Proof. 

Let G admit an H-covering consisting of exactly t subgraphs isomorphic to H and let $\psi :V\left(G\right)\to \{1,2,\dots ,k\}$ be an H-irregular vertex k-labeling of G, where $k=\mathrm{vhs}(G,H)$. Extend this labeling to the labeling of $G+F$ as follows.

Since $G+F$ contains exactly t subgraphs isomorphic to $H+F$, and each such subgraph must include all vertices of F, it follows that the contribution of labels from vertices in $V\left(F\right)$ is identical across all $(H+F)$-weights. Therefore, without loss of generality, we may assign the label 1 to each vertex in $V\left(F\right)$. In this way, the distinguishing power of the labeling relies solely on the labels of vertices in G, implying that $\mathrm{vhs}(G+F,H+F)\le k$.

Assume, for the sake of contradiction, that $\mathrm{vhs}(G+F,H+F)=k_0<k$. Then, by restricting the corresponding $(H+F)$-irregular vertex $k_0$-labeling of $G+F$ to the subgraph G, we obtain an H-irregular vertex $k_0$-labeling of G, contradicting the minimality of $k=\mathrm{vhs}(G,H)$. Hence, $\mathrm{vhs}(G+F,H+F)=k$.

If, in addition, there exists an $(H+F)$-irregular vertex $\mathrm{vhs}(G,H)$-labeling of $G+F$, where the $(H+F)$-weights form a sequence of consecutive integers, then Theorem 6 implies $\mathrm{mvhs}(G,H)=\mathrm{mvhs}(G+F,H+F)$, as claimed. □

Since the fan graph $F_n$ is isomorphic to the join $P_n+K_1$, Theorem 7 combined with Corollary 1 immediately yields the following result.

Corollary 3.

Let $F_n$, $n\ge 2$, be a fan graph on $n+1$ vertices and let m be a positive integer such that $2\le m\le n$. Then,

$$\mathrm{mvhs}(F_n,F_m)=⌈\frac{n}{m}⌉.$$

The ladder $L_n$, where $n\ge 2$, is defined as the Cartesian product of paths $P_n$ and $P_2$. In the proof of Theorem 4, the existence of an $L_m$-irregular edge $⌈\frac{2m+n-2}{3m-2}⌉$-labeling and an $L_m$-irregular vertex $⌈\frac{m+n}{2m}⌉$-labeling of the ladder $L_n$ is established. In both cases, the corresponding $L_m$-weights form a sequence of consecutive integers. With respect to Theorems 5 and 6, we have the following result:

Corollary 4.

Let $L_n\cong P_n\square P_2$, $n\ge 2$, be a ladder and let m be a positive integer such that $2\le m\le n$. Then,

$$\mathrm{mehs}(L_n,L_m)=⌈\frac{2m+n-2}{3m-2}⌉and\mathrm{mvhs}(L_n,L_m)=⌈\frac{m+n}{2m}⌉.$$

The previous corollaries confirm that the lower bounds of the modular edge H-irregularity strength and the modular vertex H-irregularity strength are tight.

3. Friendship Graphs and Ladder-Wing Graphs

A friendship graph $f_n$, where $n\ge 1$, consists of n triangles sharing a common central vertex and is otherwise independent. Let w denote the central vertex and for the ith triangle, $1\le i\le n$, let $u_i$ and $v_i$ denote the other two vertices. The friendship graph $f_n$ contains $2n+1$ vertices and $3n$ edges $u_i{v}_i$, $u_{i}w$ and $v_{i}w$ for $1\le i\le n$.

The following theorem provides the exact values of the modular edge H-irregularity strength and the modular vertex H-irregularity strength of the friendship graph $f_n$, where the subgraph H is isomorphic to $f_m$, $1\le m\le n$.

Theorem 8.

Let $f_n$, $n\ge 1$, be a friendship graph on $2n+1$ vertices and let m be a positive integer such that $1\le m\le n$. Then,

$$\mathrm{mehs}(f_n,f_m)=⌈\frac{2m+n}{3m}⌉and\mathrm{mvhs}(f_n,f_m)=⌈\frac{m+n}{2m}⌉.$$

Proof. 

Clearly, for every positive integer m such that $1\le m\le n$, the friendship graph $f_n$ on $2n+1$ vertices admits an $f_m$-covering consisting of exactly $t=n-m+1$ subgraphs isomorphic to $f_m$.

We first consider the case of edge labeling. With respect to (1), we have that $\mathrm{ehs}(f_n,f_m)\ge ⌈\frac{2m+n}{3m}⌉$. To show that $⌈\frac{2m+n}{3m}⌉$ is also an upper bound of the edge $f_m$-irregularity strength, let us define an edge labeling $\phi$ as follows:

$$\begin{array}{cccc}\hfill \phi \left(u_i{v}_i\right)& =⌈\frac{2m+i}{3m}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n,\hfill \\ \hfill \phi \left(u_{i}w\right)& =⌈\frac{i}{3m}⌉,\hfill & & \mathrm{for}i=1,2,\dots ,n,\hfill \\ \hfill \phi \left(v_{i}w\right)& =⌈\frac{m+i}{3m}⌉,\hfill & & \mathrm{for}i=1,2,\dots ,n.\hfill \end{array}$$

Evidently, $\phi$ is a $⌈\frac{2m+n}{3m}⌉$-labeling. Let us now compute the $f_m$-weight of the subgraph $f_m^j$, $j=1,2,\dots ,n-m+1$, under the labeling $\phi$:

$$\begin{array}{cc}\hfill w{t}_{\phi }\left(f_m^j\right)=& \sum _{e\in E\left(f_m^j\right)}\phi \left(e\right)=\sum _{i=j}^{m+j-1}\left[\phi \left(u_i{v}_i\right)+\phi \left(u_{i}w\right)+\phi \left(v_{i}w\right)\right]=\phi \left(u_j{v}_j\right)+\phi \left(u_{j}w\right)+\phi \left(v_{j}w\right)\hfill \\ \hfill & +\sum _{i=j+1}^{m+j-1}\left[\phi \left(u_i{v}_i\right)+\phi \left(u_{i}w\right)+\phi \left(v_{i}w\right)\right]=⌈\frac{2m+j}{3m}⌉+⌈\frac{j}{3m}⌉+⌈\frac{m+j}{3m}⌉\hfill \\ & +\sum _{i=j+1}^{m+j-1}\left[\phi \left(u_i{v}_i\right)+\phi \left(u_{i}w\right)+\phi \left(v_{i}w\right)\right].\hfill \end{array}$$

Similarly, for the weight of the next subgraph $f_m^{j+1}$, $j=1,2,\dots ,n-m$, we have

$$\begin{array}{cc}\hfill w{t}_{\phi }\left(f_m^{j+1}\right)=& \sum _{e\in E\left(f_m^{j+1}\right)}\phi \left(e\right)=\sum _{i=j+1}^{m+j}\left[\phi \left(u_i{v}_i\right)+\phi \left(u_{i}w\right)+\phi \left(v_{i}w\right)\right]=\phi \left(u_{m+j}{v}_{m+j}\right)+\phi \left(u_{m+j}w\right)\hfill \\ & +\phi \left(v_{m+j}w\right)+\sum _{i=j+1}^{m+j-1}\left[\phi \left(u_i{v}_i\right)+\phi \left(u_{i}w\right)+\phi \left(v_{i}w\right)\right]=⌈\frac{2m+m+j}{3m}⌉+⌈\frac{m+j}{3m}⌉\hfill \\ & +⌈\frac{m+m+j}{3m}⌉+\sum _{i=j+1}^{m+j-1}\left[\phi \left(u_i{v}_i\right)+\phi \left(u_{i}w\right)+\phi \left(v_{i}w\right)\right]=1+w{t}_{\phi }\left(f_m^j\right).\hfill \end{array}$$

Thus, the $f_m$-weights under the labeling $\phi$ form the set of consecutive integers $\{3m,3m+1,\dots ,n+2m\}$. It follows that $\phi$ is an $f_m$-irregular edge $⌈\frac{2m+n}{3m}⌉$-labeling. Combining this with Theorem 5, we conclude that $\mathrm{ehs}(f_n,f_m)=\mathrm{mehs}(f_n,f_m)=⌈\frac{2m+n}{3m}⌉$.

Now, we turn our attention to vertex labeling. From Inequality (2), we obtain the lower bound $\mathrm{vhs}(f_n,f_m)\ge ⌈\frac{m+n+1}{2m+1}⌉$. On the other hand, under any vertex labeling $\psi$ of $f_n$, the smallest $f_m$-weight is at least $2m+\psi \left(w\right)$, and therefore, the largest $f_m$-weight is at least $2m+\psi \left(w\right)+t-1=n+m+\psi \left(w\right)$. Assuming that $\mathrm{vhs}(f_n,f_m)=k$, it follows that the largest $f_m$-weight is at most $2mk+\psi \left(w\right)$. Combining these inequalities yields $n+m+\psi \left(w\right)\le 2mk+\psi \left(w\right)$, which simplifies to $k\ge ⌈\frac{n+m}{2m}⌉$. To show that $⌈\frac{n+m}{2m}⌉$ is also an upper bound for the vertex $f_m$-irregularity strength of $f_n$, we define a vertex labeling $\psi$ as follows:

$$\begin{array}{cccc}\hfill \psi \left(u_i\right)& =⌈\frac{m+i}{2m}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n,\hfill \\ \hfill \psi \left(v_i\right)& =⌈\frac{i}{2m}⌉,\hfill & & \mathrm{for}i=1,2,\dots ,n.\hfill \end{array}$$

Without loss of generality, we may assign the central vertex w the value $\psi \left(w\right)=1$. Clearly, $\psi \left(v_i\right)\le \psi \left(u_i\right)\le ⌈\frac{n+m}{2m}⌉$, and hence, $\psi$ is a vertex $⌈\frac{n+m}{2m}⌉$-labeling. Next, we evaluate the corresponding $f_m$-weights. For each $j=1,2,\dots ,n-m+1$, we have

$$\begin{array}{cc}\hfill w{t}_{\psi }\left(f_m^j\right)=& \sum _{v\in V\left(f_m^j\right)}\psi \left(v\right)=\sum _{i=j}^{m+j-1}\left[\psi \left(u_i\right)+\psi \left(v_i\right)\right]+\psi \left(w\right)=\psi \left(u_j\right)+\psi \left(v_j\right)+\psi \left(w\right)\hfill \\ \hfill & +\sum _{i=j+1}^{m+j-1}\left[\psi \left(u_i\right)+\psi \left(v_i\right)\right]=⌈\frac{m+j}{2m}⌉+⌈\frac{j}{2m}⌉+1+\sum _{i=j+1}^{m+j-1}\left[\psi \left(u_i\right)+\psi \left(v_i\right)\right].\hfill \end{array}$$

Similarly, for $j=1,2,\dots ,n-m$, we obtain

$$\begin{array}{cc}\hfill w{t}_{\psi }\left(f_m^{j+1}\right)=& \sum _{v\in V\left(f_m^{j+1}\right)}\psi \left(v\right)=\sum _{i=j+1}^{m+j-1}\left[\psi \left(u_i\right)+\psi \left(v_i\right)\right]+\psi \left(w\right)+\psi \left(u_{m+j}\right)+\psi \left(v_{m+j}\right)\hfill \\ \hfill & =\sum _{i=j+1}^{m+j-1}\left[\psi \left(u_i\right)+\psi \left(v_i\right)\right]+1+⌈\frac{2m+j}{2m}⌉+⌈\frac{m+j}{2m}⌉=1+w{t}_{\psi }\left(f_m^j\right).\hfill \end{array}$$

Thus, $\psi$ is an $f_m$-irregular vertex $⌈\frac{m+n}{2m}⌉$-labeling. Moreover, since the weights of subgraphs $f_m^j$ for $j=1,2,\dots ,n-m+1$ are not only distinct but form a sequence of consecutive integers $\{2m+1,2m+2,\dots ,n+m+1\}$, it follows from Theorem 6 that $\mathrm{vhs}(f_n,f_m)=\mathrm{mvhs}(f_n,f_m)=⌈\frac{m+n}{2m}⌉$. This completes the proof. □

The ladder-wing graph $L{W}_n$, for $n\ge 2$, is obtained by joining all vertices of a ladder $L_n$ to an additional vertex; let us denote it as w. In other words, $L{W}_n=L_n+K_1$. The following theorem determines the exact values of the modular edge H-irregularity strength and the modular vertex H-irregularity strength for the ladder-wing graph $L{W}_n$, where the subgraph H is isomorphic to $L{W}_m$, $2\le m\le n$.

Theorem 9.

Let $L{W}_n$, $n\ge 2$, be a ladder-wing graph on $2n+1$ vertices and let m be a positive integer such that $2\le m\le n$. Then,

$$\mathrm{mehs}(L{W}_n,L{W}_m)=⌈\frac{4m+n-2}{5m-2}⌉and\mathrm{mvhs}(L{W}_n,L{W}_m)=⌈\frac{m+n}{2m}⌉.$$

Proof. 

Let $L{W}_n$ be a ladder-wing graph on $2n+1$ vertices with vertex set $V\left(L{W}_n\right)=\{u_i,v_i:i=1,2,\dots ,n\}\cup \left\{w\right\}$ and edge set $E\left(L{W}_n\right)=\{u_i{u}_{i+1},v_i{v}_{i+1}:i=1,2,\dots ,n-1\}\cup \{u_i{v}_i,u_{i}w,v_{i}w:i=1,2,\dots ,n\}$. Clearly, for every m, $2\le m\le n$, the ladder-wing graph $L{W}_n$ admits a $L{W}_m$-covering consisting of exactly $t=n-m+1$ subgraphs isomorphic to $L{W}_m$.

We first deal with the edge labeling. According to (1), we have $\mathrm{ehs}(L{W}_n,L{W}_m)\ge ⌈\frac{4m+n-2}{5m-2}⌉$. To prove the equality, it suffices to construct an edge $⌈\frac{4m+n-2}{5m-2}⌉$-labeling of $L{W}_n$ that is $L{W}_m$-irregular.

For $n\ge 2$, define an edge labeling $\phi$ on $L{W}_n$ as follows:

$$\begin{array}{cccc}\hfill \phi \left(u_i{u}_{i+1}\right)& =⌈\frac{3m+i}{5m-2}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n-1,\hfill \\ \hfill \phi \left(v_i{v}_{i+1}\right)& =⌈\frac{4m-1+i}{5m-2}⌉,\hfill & & \mathrm{for}i=1,2,\dots ,n-1,\hfill \\ \hfill \phi \left(u_i{v}_i\right)& =⌈\frac{2m+i}{5m-2}⌉,\hfill & & \mathrm{for}i=1,2,\dots ,n,\hfill \\ \hfill \phi \left(u_{i}w\right)& =⌈\frac{i}{5m-2}⌉,\hfill & & \mathrm{for}i=1,2,\dots ,n,\hfill \\ \hfill \phi \left(v_{i}w\right)& =⌈\frac{m+i}{5m-2}⌉,\hfill & & \mathrm{for}i=1,2,\dots ,n.\hfill \end{array}$$

All values are clearly bounded above by $⌈\frac{4m+n-2}{5m-2}⌉$; so, $\phi$ is an edge $⌈\frac{4m+n-2}{5m-2}⌉$-labeling.

Now, consider the $L{W}_m$-subgraphs $L{W}_m^j$ for $j=1,2,\dots ,n-m+1$. The weight of each such subgraph is

$$\begin{array}{cc}\hfill w{t}_{\phi }\left(L{W}_m^j\right)=& \sum _{e\in E\left(L{W}_m^j\right)}\phi \left(e\right)=\sum _{i=j}^{m+j-1}\left[\phi \left(u_i{v}_i\right)+\phi \left(u_{i}w\right)+\phi \left(v_{i}w\right)\right]\hfill \\ \hfill & +\sum _{i=j}^{m+j-2}\left[\phi \left(u_i{u}_{i+1}\right)+\phi \left(v_i{v}_{i+1}\right)\right]=\phi \left(u_j{v}_j\right)+\phi \left(u_{j}w\right)+\phi \left(v_{j}w\right)+\phi \left(u_j{u}_{j+1}\right)\hfill \\ \hfill & +\phi \left(v_j{v}_{j+1}\right)+\sum _{i=j+1}^{m+j-1}\left[\phi \left(u_i{v}_i\right)+\phi \left(u_{i}w\right)+\phi \left(v_{i}w\right)\right]\hfill \\ \hfill & +\sum _{i=j+1}^{m+j-2}\left[\phi \left(u_i{u}_{i+1}\right)+\phi \left(v_i{v}_{i+1}\right)\right]\hfill \\ \hfill =& ⌈\frac{2m+j}{5m-2}⌉+⌈\frac{j}{5m-2}⌉+⌈\frac{m+j}{5m-2}⌉+⌈\frac{3m+j}{5m-2}⌉+⌈\frac{4m-1+j}{5m-2}⌉\hfill \\ \hfill & +\sum _{i=j+1}^{m+j-1}\left[\phi \left(u_i{v}_i\right)+\phi \left(u_{i}w\right)+\phi \left(v_{i}w\right)\right]+\sum _{i=j+1}^{m+j-2}\left[\phi \left(u_i{u}_{i+1}\right)+\phi \left(v_i{v}_{i+1}\right)\right].\hfill \end{array}$$

For the next subgraph $L{W}_m^{j+1}$, $j=1,2,\dots ,n-m$, the analogous weight is

$$\begin{array}{cc}\hfill w{t}_{\phi }\left(L{W}_m^{j+1}\right)=& \sum _{e\in E\left(L{W}_m^{j+1}\right)}\phi \left(e\right)=\sum _{i=j+1}^{m+j}\left[\phi \left(u_i{v}_i\right)+\phi \left(u_{i}w\right)+\phi \left(v_{i}w\right)\right]\hfill \\ \hfill & +\sum _{i=j+1}^{m+j-1}\left[\phi \left(u_i{u}_{i+1}\right)+\phi \left(v_i{v}_{i+1}\right)\right]=\phi \left(u_{m+j}{v}_{m+j}\right)+\phi \left(u_{m+j}w\right)+\phi \left(v_{m+j}w\right)\hfill \\ \hfill & +\phi \left(u_{m+j-1}{u}_{m+j}\right)+\phi \left(v_{m+j-1}{v}_{m+j}\right)+\sum _{i=j+1}^{m+j-1}\left[\phi \left(u_i{v}_i\right)+\phi \left(u_{i}w\right)+\phi \left(v_{i}w\right)\right]\hfill \\ \hfill & +\sum _{i=j+1}^{m+j-2}\left[\phi \left(u_i{u}_{i+1}\right)+\phi \left(v_i{v}_{i+1}\right)\right]=⌈\frac{3m+j}{5m-2}⌉+⌈\frac{m+j}{5m-2}⌉+⌈\frac{2m+j}{5m-2}⌉\hfill \\ \hfill & +⌈\frac{4m+j-1}{5m-2}⌉+⌈\frac{5m-2+j}{5m-2}⌉+\sum _{i=j+1}^{m+j-1}\left[\phi \left(u_i{v}_i\right)+\phi \left(u_{i}w\right)+\phi \left(v_{i}w\right)\right]\hfill \\ \hfill & +\sum _{i=j+1}^{m+j-2}\left[\phi \left(u_i{u}_{i+1}\right)+\phi \left(v_i{v}_{i+1}\right)\right]=1+w{t}_{\phi }\left(L{W}_m^j\right).\hfill \end{array}$$

Thus, the $L{W}_m$-weights form a sequence of consecutive integers starting at $5m-2$ and ending at $4m+n-2$, i.e., $\{5m-2,5m-1,\dots ,4m+n-2\}$. By Theorem 5, it follows that $\mathrm{ehs}(L{W}_n,L{W}_m)=\mathrm{mehs}(L{W}_n,L{W}_m)=⌈\frac{4m+n-2}{5m-2}⌉$.

Now, consider the vertex case. By definition, the ladder-wing graph is obtained as the join of a ladder and an isolated vertex, that is, $L{W}_n=L_n+K_1$. Therefore, the desired result $\mathrm{vhs}(L{W}_n,L{W}_m)=\mathrm{mvhs}(L{W}_n,L{W}_m)=⌈\frac{m+n}{2m}⌉$ follows immediately by combining Theorem 7 and Corollary 4. □

In the next result, we consider a graph obtained from the ladder graph by inserting both diagonals into every four-sided face of the ladder. We denote this graph by $R_n$. Alternatively, $R_n$ can be defined as the strong product of the path on n vertices and the path on two vertices. Let the vertex set and edge set of $R_n$ be given by $V\left(R_n\right)=\{u_i,v_i:i=1,2,\dots ,n\}$ and $E\left(R_n\right)=\{u_i{u}_{i+1},v_i{v}_{i+1},u_i{v}_{i+1},u_{i+1}{v}_i:i=1,2,\dots ,n-1\}\cup \{u_i{v}_i:i=1,2,\dots ,n\}$. The following theorem gives the exact values of the modular edge H-irregularity strength and the modular vertex H-irregularity strength for the graph $R_n$, where the subgraph H is isomorphic to $R_m$, with $2\le m\le n$.

Theorem 10.

Let $R_n$, $n\ge 2$, be the strong product of the path on n vertices and the path on two vertices and let m be a positive integer such that $2\le m\le n$. Then,

$$\mathrm{mehs}(R_n,R_m)=⌈\frac{4m+n-4}{5m-4}⌉and\mathrm{mvhs}(R_n,R_m)=⌈\frac{m+n}{2m}⌉.$$

Proof. 

One can observe that the ladder $R_n\cong P_n⊠P_2$, $n\ge 2$, admits an $R_m$-covering consisting of exactly $t=n-m+1$ subgraphs isomorphic to $R_m$, where m is a positive integer satisfying $2\le m\le n$. From Inequality (1), it follows that $\mathrm{ehs}(R_n,R_m)\ge ⌈\frac{4m+n-4}{5m-4}⌉$. Set $k=⌈\frac{4m+n-4}{5m-4}⌉$. To show that k is also an upper bound for the edge $R_m$-irregularity strength of $R_n$, we define an edge labeling $\phi$ as follows:

$$\begin{array}{cccc}\hfill \phi \left(u_i{u}_{i+1}\right)& =⌈\frac{4m-3+i}{5m-4}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n-1,\hfill \\ \hfill \phi \left(v_i{v}_{i+1}\right)& =⌈\frac{3m-2+i}{5m-4}⌉,\hfill & & \mathrm{for}i=1,2,\dots ,n-1,\hfill \\ \hfill \phi \left(u_i{v}_i\right)& =⌈\frac{2m-2+i}{5m-4}⌉,\hfill & & \mathrm{for}i=1,2,\dots ,n,\hfill \\ \hfill \phi \left(u_i{v}_{i+1}\right)& =⌈\frac{m-1+i}{5m-4}⌉,\hfill & & \mathrm{for}i=1,2,\dots ,n-1,\hfill \\ \hfill \phi \left(u_{i+1}{v}_i\right)& =⌈\frac{i}{5m-4}⌉,\hfill & & \mathrm{for}i=1,2,\dots ,n-1.\hfill \end{array}$$

It is straightforward to verify that all edge labels used by $\phi$ are at most k. For the $R_m$-weight of the subgraph $R_m^j$, $j=1,2,\dots ,n-m+1$, under the labeling $\phi$, we obtain

$$\begin{array}{cc}\hfill w{t}_{\phi }\left(R_m^j\right)=& \sum _{e\in E\left(R_m^j\right)}\phi \left(e\right)=\sum _{i=j}^{m+j-2}\left[\phi \left(u_i{u}_{i+1}\right)+\phi \left(v_i{v}_{i+1}\right)+\phi \left(u_i{v}_{i+1}\right)+\phi \left(u_{i+1}{v}_i\right)\right]\hfill \\ \hfill & +\sum _{i=j}^{m+j-1}\phi \left(u_i{v}_i\right)=\phi \left(u_j{u}_{j+1}\right)+\phi \left(v_j{v}_{j+1}\right)+\phi \left(u_j{v}_{j+1}\right)+\phi \left(u_{j+1}{v}_j\right)+\phi \left(u_j{v}_j\right)\hfill \\ \hfill & +\sum _{i=j+1}^{m+j-2}\left[\phi \left(u_i{u}_{i+1}\right)+\phi \left(v_i{v}_{i+1}\right)+\phi \left(u_i{v}_{i+1}\right)+\phi \left(u_{i+1}{v}_i\right)\right]+\sum _{i=j+1}^{m+j-1}\phi \left(u_i{v}_i\right)\hfill \\ \hfill =& ⌈\frac{4m-3+j}{5m-4}⌉+⌈\frac{3m-2+j}{5m-4}⌉+⌈\frac{m-1+j}{5m-4}⌉+⌈\frac{j}{5m-4}⌉+⌈\frac{2m-2+j}{5m-4}⌉\hfill \\ \hfill & +\sum _{i=j+1}^{m+j-2}\left[\phi \left(u_i{u}_{i+1}\right)+\phi \left(v_i{v}_{i+1}\right)+\phi \left(u_i{v}_{i+1}\right)+\phi \left(u_{i+1}{v}_i\right)\right]+\sum _{i=j+1}^{m+j-1}\phi \left(u_i{v}_i\right).\hfill \end{array}$$

For the $R_m$-weight of the subgraph $R_m^{j+1}$, $j=1,2,\dots ,n-m$, we have

$$\begin{array}{cc}\hfill w{t}_{\phi }\left(R_m^{j+1}\right)=& \sum _{e\in E\left(R_m^{j+1}\right)}\phi \left(e\right)=\sum _{i=j+1}^{m+j-1}\left[\phi \left(u_i{u}_{i+1}\right)+\phi \left(v_i{v}_{i+1}\right)+\phi \left(u_i{v}_{i+1}\right)+\phi \left(u_{i+1}{v}_i\right)\right]\hfill \\ \hfill & +\sum _{i=j+1}^{m+j}\phi \left(u_i{v}_i\right)=\sum _{i=j+1}^{m+j-2}\left[\phi \left(u_i{u}_{i+1}\right)+\phi \left(v_i{v}_{i+1}\right)+\phi \left(u_i{v}_{i+1}\right)+\phi \left(u_{i+1}{v}_i\right)\right]\hfill \\ \hfill & +\phi \left(u_{m+j-1}{u}_{m+j}\right)+\phi \left(v_{m+j-1}{v}_{m+j}\right)+\phi \left(u_{m+j-1}{v}_{m+j}\right)+\phi \left(u_{m+j}{v}_{m+j-1}\right)\hfill \\ \hfill & +\sum _{i=j+1}^{m+j-1}\phi \left(u_i{v}_i\right)+\phi \left(u_{m+j}{v}_{m+j}\right)=\sum _{i=j+1}^{m+j-2}\left[\phi \left(u_i{u}_{i+1}\right)+\phi \left(v_i{v}_{i+1}\right)+\phi \left(u_i{v}_{i+1}\right)\right.\hfill \\ \hfill & \left.+\phi \left(u_{i+1}{v}_i\right)\right]+⌈\frac{5m-4+j}{5m-4}⌉+⌈\frac{4m-3+j}{5m-4}⌉+⌈\frac{2m-2+j}{5m-4}⌉+⌈\frac{m-1+j}{5m-4}⌉\hfill \\ \hfill & +\sum _{i=j+1}^{m+j-1}\phi \left(u_i{v}_i\right)+⌈\frac{3m-2+j}{5m-4}⌉=1+w{t}_{\phi }\left(R_m^j\right).\hfill \end{array}$$

This means that $R_m$-weights successively attain values $5m-4,5m-3,\dots ,4m+n-4$ and are distinct for all pairs of distinct subgraph $R_m$. The edge labeling $\phi$ thus provides an upper bound on the edge $R_m$-irregularity strength of the ladder $R_n$. Consequently, we obtain $\mathrm{ehs}(R_n,R_m)=⌈\frac{4m+n-4}{5m-4}⌉$. According to Theorem 5, we obtain that $\mathrm{ehs}(R_n,R_m)=\mathrm{mehs}(R_n,R_m)$.

Since the graphs $P_n⊠P_2$ and $P_n\square P_2$ have the same vertex set, and all corresponding subgraphs $P_m⊠P_2$ and $P_m\square P_2$ likewise have identical vertex sets, it follows from Corollary 4 that $\mathrm{mvhs}(R_n,R_m)=\mathrm{mvhs}(L_n,L_m)=⌈\frac{m+n}{2m}⌉$. This concludes the proof. □

Consider a new graph, called the ladder-wing graph and denoted by $R{W}_n$, $n\ge 2$, which is obtained by joining all vertices of the ladder $R_n\cong P_n⊠P_2$ to an additional vertex w. This graph has $2n+1$ vertices, with vertex set $V\left(R{W}_n\right)=\{u_i,v_i:i=1,2,\dots ,n\}\cup \left\{w\right\}$ and edge set $E\left(R{W}_n\right)=\{u_i{u}_{i+1},v_i{v}_{i+1},u_i{v}_{i+1},u_{i+1}{v}_i:i=1,2,\dots ,n-1\}\cup \{u_i{v}_i,u_{i}w,v_{i}w:i=1,2,\dots ,n\}$.

The following theorem provides the exact values of the modular edge H-irregularity strength and the modular vertex H-irregularity strength for ladder-wing graph $R{W}_n$, where the subgraph H is isomorphic to $R{W}_m$ with $2\le m\le n$.

Theorem 11.

Let $R{W}_n$, $n\ge 2$, be a ladder-wing graph on $2n+1$ vertices and let m be a positive integer such that $2\le m\le n$. Then,

$$\mathrm{mehs}(R{W}_n,R{W}_m)=⌈\frac{6m+n-4}{7m-4}⌉and\mathrm{mvhs}(R{W}_n,R{W}_m)=⌈\frac{m+n}{2m}⌉.$$

Proof. 

Clearly, the ladder-wing graph $R{W}_n$, $n\ge 2$, admits an $R{W}_m$-covering consisting of exactly $t=n-m+1$ subgraphs for every m such that $2\le m\le n$. According to (1), it follows that

$$\begin{array}{c}\hfill \mathrm{ehs}(R{W}_n,R{W}_m)\ge ⌈\frac{6m+n-4}{7m-4}⌉.\end{array}$$

(7)

To prove the converse inequality, we construct an edge labeling $\phi$ such that

$$\begin{array}{cccc}\hfill \phi \left(u_i{u}_{i+1}\right)& =⌈\frac{6m-3+i}{7m-4}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n-1,\hfill \\ \hfill \phi \left(v_i{v}_{i+1}\right)& =⌈\frac{5m-2+i}{7m-4}⌉,\hfill & & \mathrm{for}i=1,2,\dots ,n-1,\hfill \\ \hfill \phi \left(u_i{v}_i\right)& =⌈\frac{4m-2+i}{7m-4}⌉,\hfill & & \mathrm{for}i=1,2,\dots ,n,\hfill \\ \hfill \phi \left(u_i{v}_{i+1}\right)& =⌈\frac{3m-1+i}{7m-4}⌉,\hfill & & \mathrm{for}i=1,2,\dots ,n-1,\hfill \\ \hfill \phi \left(u_{i+1}{v}_i\right)& =⌈\frac{2m+i}{7m-4}⌉,\hfill & & \mathrm{for}i=1,2,\dots ,n-1,\hfill \\ \hfill \phi \left(u_{i}w\right)& =⌈\frac{m+i}{7m-4}⌉,\hfill & & \mathrm{for}i=1,2,\dots ,n,\hfill \\ \hfill \phi \left(v_{i}w\right)& =⌈\frac{i}{7m-4}⌉,\hfill & & \mathrm{for}i=1,2,\dots ,n.\hfill \end{array}$$

It is straightforward to verify that under the previously defined labeling, all edge labels are at most $⌈\frac{6m+n-4}{7m-4}⌉$. For the $R{W}_m$-weight of the subgraph $R{W}_m^j$, $j=1,2,\dots ,n-m+1$, we obtain

$$\begin{array}{cc}\hfill w{t}_{\phi }\left(R{W}_m^j\right)=& \sum _{e\in E\left(R{W}_m^j\right)}\phi \left(e\right)=\sum _{i=j}^{m+j-2}\left[\phi \left(u_i{u}_{i+1}\right)+\phi \left(v_i{v}_{i+1}\right)+\phi \left(u_i{v}_{i+1}\right)+\phi \left(u_{i+1}{v}_i\right)\right]\hfill \\ \hfill & +\sum _{i=j}^{m+j-1}\left[\phi \left(u_i{v}_i\right)+\phi \left(u_{i}w\right)+\phi \left(v_{i}w\right)\right]=\phi \left(u_j{u}_{j+1}\right)+\phi \left(v_j{v}_{j+1}\right)+\phi \left(u_j{v}_{j+1}\right)\hfill \\ \hfill & +\phi \left(u_{j+1}{v}_j\right)+\sum _{i=j+1}^{m+j-2}\left[\phi \left(u_i{u}_{i+1}\right)+\phi \left(v_i{v}_{i+1}\right)+\phi \left(u_i{v}_{i+1}\right)+\phi \left(u_{i+1}{v}_i\right)\right]\hfill \\ \hfill & +\phi \left(u_j{v}_j\right)+\phi \left(u_{j}w\right)+\phi \left(v_{j}w\right)+\sum _{i=j+1}^{m+j-1}\left[\phi \left(u_i{v}_i\right)+\phi \left(u_{i}w\right)+\phi \left(v_{i}w\right)\right]\hfill \\ \hfill =& ⌈\frac{6m-3+j}{7m-4}⌉+⌈\frac{5m-2+j}{7m-4}⌉+⌈\frac{3m-1+j}{7m-4}⌉+⌈\frac{2m+j}{7m-4}⌉\hfill \\ \hfill & +\sum _{i=j+1}^{m+j-2}\left[\phi \left(u_i{u}_{i+1}\right)+\phi \left(v_i{v}_{i+1}\right)+\phi \left(u_i{v}_{i+1}\right)+\phi \left(u_{i+1}{v}_i\right)\right]+⌈\frac{4m-2+j}{7m-4}⌉\hfill \\ \hfill & +⌈\frac{m+j}{7m-4}⌉+⌈\frac{j}{7m-4}⌉+\sum _{i=j+1}^{m+j-1}\left[\phi \left(u_i{v}_i\right)+\phi \left(u_{i}w\right)+\phi \left(v_{i}w\right)\right].\hfill \end{array}$$

For the $R{W}_m$-weight of the subgraph $R{W}_m^{j+1}$, $j=1,2,\dots ,n-m$, we obtain

$$\begin{array}{cc}\hfill w{t}_{\phi }\left(R{W}_m^{j+1}\right)=& \sum _{e\in E\left(R{W}_m^{j+1}\right)}\phi \left(e\right)=\sum _{i=j+1}^{m+j-1}\left[\phi \left(u_i{u}_{i+1}\right)+\phi \left(v_i{v}_{i+1}\right)+\phi \left(u_i{v}_{i+1}\right)+\phi \left(u_{i+1}{v}_i\right)\right]\hfill \\ \hfill & +\sum _{i=j+1}^{m+j}\left[\phi \left(u_i{v}_i\right)+\phi \left(u_{i}w\right)+\phi \left(v_{i}w\right)\right]=\sum _{i=j+1}^{m+j-2}\left[\phi \left(u_i{u}_{i+1}\right)+\phi \left(v_i{v}_{i+1}\right)\right.\hfill \\ \hfill & \left.+\phi \left(u_i{v}_{i+1}\right)+\phi \left(u_{i+1}{v}_i\right)\right]+\phi \left(u_{m+j-1}{u}_{m+j}\right)+\phi \left(v_{m+j-1}{v}_{m+j}\right)\hfill \\ \hfill & +\phi \left(u_{m+j-1}{v}_{m+j}\right)+\phi \left(u_{m+j}{v}_{m+j-1}\right)+\sum _{i=j+1}^{m+j-1}\left[\phi \left(u_i{v}_i\right)+\phi \left(u_{i}w\right)+\phi \left(v_{i}w\right)\right]\hfill \\ \hfill & +\phi \left(u_{m+j}{v}_{m+j}\right)+\phi \left(u_{m+j}w\right)+\phi \left(v_{m+j}w\right)=\sum _{i=j+1}^{m+j-2}\left[\phi \left(u_i{u}_{i+1}\right)+\phi \left(v_i{v}_{i+1}\right)\right.\hfill \\ \hfill & \left.+\phi \left(u_i{v}_{i+1}\right)+\phi \left(u_{i+1}{v}_i\right)\right]+⌈\frac{7m-4+j}{7m-4}⌉+⌈\frac{6m-3+j}{7m-4}⌉+⌈\frac{4m-2+j}{7m-4}⌉\hfill \\ \hfill & +⌈\frac{3m-1+j}{7m-4}⌉+\sum _{i=j+1}^{m+j-1}\left[\phi \left(u_i{v}_i\right)+\phi \left(u_{i}w\right)+\phi \left(v_{i}w\right)\right]+⌈\frac{5m-2+j}{7m-4}⌉\hfill \\ \hfill & +⌈\frac{2m+j}{7m-4}⌉+⌈\frac{m+j}{7m-4}⌉=1+w{t}_{\phi }\left(R{W}_m^j\right).\hfill \end{array}$$

Observe that $R{W}_m$-weights successively receive values $7m-4,7m-3,\dots ,6m+n-4$. This implies that the weights are distinct for all pairs of distinct subgraphs isomorphic to $R_m$, and therefore, edge labeling $\phi$ is a desired $R{W}_m$-irregular edge $⌈\frac{6m+n-4}{7m-4}⌉$-labeling. Hence, this labeling provides an upper bound on $\mathrm{ehs}(R{W}_n,R{W}_m)$. Combined with the lower bounds established in (7), we conclude that $\mathrm{ehs}(R{W}_n,R{W}_m)=⌈\frac{6m+n-4}{7m-4}⌉$. The desired result then follows directly from Theorem 5.

Since the graphs $L{W}_n$ and $R{W}_n$ have the same vertex set, and each corresponding pair of subgraphs $L{W}_m$ and $R{W}_m$ likewise have identical vertex sets, it follows from Theorem 9 that $\mathrm{mvhs}(L{W}_n,L{W}_m)=\mathrm{mvhs}(R{W}_n,R{W}_m)=⌈\frac{m+n}{2m}⌉$. This completes the proof. □

4. Conclusions

In this paper, we introduce two novel graph invariants: the modular edge H-irregularity strength, denoted $\mathrm{mehs}(G,H)$, and the modular vertex H-irregularity strength, denoted $\mathrm{mvhs}(G,H)$. These are natural extensions of existing irregularity strength concepts. We establish general lower bounds and determine exact values for several graph families, including fan graphs, friendship graphs, ladders, and ladder-wing graphs. In each case, the subgraphs H are isomorphic to graphs of the same type but on a smaller number of vertices. These results confirm the sharpness of the proposed bounds and demonstrate the utility of modular labelings in distinguishing subgraph structures.

It may be of interest to determine the exact values of these invariants in cases where G and H are not of the same type. In the examples presented, the subgraphs H are, in some sense, linearly ordered. It would be challenging to explore cases where the H-covering of G has a more complex structure. A simple example is the case where a cycle $C_n$ is covered by paths $P_m$ for $2\le m\le n$. Considering the two extremal cases, covering the cycle $C_n$ by $P_2$ and $P_n$, we trivially obtain $\mathrm{mehs}(C_n,P_2)=\mathrm{mehs}(C_n,P_n)=n$. This follows from the fact that the $P_2$-weights correspond to the label of a single edge, which must be unique since all weights must differ. For $P_n$, the weights equal the sum of all but one edge label on $C_n$. For $m=3$, we have that $\mathrm{mehs}(C_n,P_3)$ equals the modular vertex-irregularity strength $\mathrm{ms}\left(C_n\right)$, as defined by Bača et al. in [16]. This is because the weight of each path $P_3$ in $C_n$ essentially corresponds to the vertex weight of the vertex of degree 2 in the path. This observation can be trivially generalized to any r-regular graph G, where $\mathrm{mehs}(G,K_{1,r})=\mathrm{ms}\left(G\right)$. Moreover, for regular graphs, the concept of $\mathrm{vhs}(G,K_{1,r})$ is closely related to the inclusive distance vertex irregularity strength introduced by Bača et al. [19].

Future work may focus on determining $\mathrm{mehs}(G,H)$ and $\mathrm{mvhs}(G,H)$ for other graph families, as well as exploring algorithmic aspects and investigating connections with other graph operations or labeling parameters.

All of these concepts open new avenues for studying labeling parameters that induce not only distinct subgraph weights, but weights that are distinct modulo the number of subgraphs. This implies that subgraphs can be uniquely identified by their weights, which may serve as a foundation for potential applications—an exciting direction we hope will be explored in future research.