Modular Total H-Irregularity Strength of Graphs

1. Introduction

Let $G=(V,E)$ be a simple graph with vertex set $V\left(G\right)$ and edge set $E\left(G\right)$. A labeling of a graph is a map that carries graph elements to the positive or non-negative integers. If the domain is the vertex (edge) set alone or the set of all vertices and edges, the labelings are respectively called vertex (edge) labelings or total labelings.

For a graph G and a positive integer k, Bača et al. [1] introduced a total labeling $\psi :V\left(G\right)\cup E\left(G\right)\to \{1,2,\dots ,k\}$ called an edge irregular total k-labeling having the property that distinct edges of G have distinct total edge weights, where the total edge weight of an edge $xy$ is defined as $w{t}_{\psi }\left(xy\right)=\psi \left(x\right)+\psi \left(xy\right)+\psi \left(y\right)$. The minimum k for which G has an edge irregular total k-labeling is called the total edge irregularity strength of G, abbreviated as $\mathrm{tes}\left(G\right)$.

A lower bound on the total edge irregularity strength of a graph G of maximum degree $\Delta \left(G\right)$ is given in [1] in the following form

$$\mathrm{tes}\left(G\right)\ge max\left\{⌈\frac{\left|E\right(G\left)\right|+2}{3}⌉,⌈\frac{\Delta \left(G\right)+1}{2}⌉\right\}.$$

(1)

Ivančo and Jendrol’ [2] posed a conjecture that for all graphs different from $K_5$ the total edge irregularity strength is exactly the maximum of these two values, namely, $⌈\left(\right|E\left(G\right)|+2)/3⌉$ and $⌈\left(\Delta \right(G)+1)/2⌉$. This conjecture has been verified for complete graphs and complete bipartite graphs in [3] and [4], for large dense graphs with $\left(\right|E\left(G\right)|+2)/3\le \left(\Delta \right(G)+1)/2$ in [5], for generalized Petersen graphs in [6], for generalized prisms in [7], for the categorical product of two cycles in [8] and for corona product of a path with certain graphs in [9].

In [1], the authors also defined a vertex irregular total k-labeling. A total labeling $\psi :V\left(G\right)\cup E\left(G\right)\to \{1,2,\dots ,k\}$ is called a vertex irregular total k-labeling if $w{t}_{\psi }\left(x\right)\ne w{t}_{\psi }\left(y\right)$, for every pair of distinct vertices x and y of G, where the total vertex weight of a vertex x is defined as

$$w{t}_{\psi }\left(x\right)=\psi \left(x\right)+\sum _{xy\in E\left(G\right)}\psi \left(xy\right).$$

The minimum k for which G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G, denoted by $\mathrm{tvs}\left(G\right)$.

The following bounds were proved in [1] in terms of the maximum degree $\Delta \left(G\right)$ and minimum degree $\delta \left(G\right)$ of G:

$$\begin{array}{c}\hfill ⌈\frac{\left|V\right(G\left)\right|+\delta \left(G\right)}{\Delta \left(G\right)+1}⌉\le \mathrm{tvs}\left(G\right)\le \left|V\left(G\right)\right|+\Delta \left(G\right)-2\delta \left(G\right)+1.\end{array}$$

(2)

Moreover, for graphs each of whose components has at least three vertices, an upper bound is given by

$$\begin{array}{c}\hfill \mathrm{tvs}\left(G\right)\le \left|V\left(G\right)\right|-1-⌊\frac{\left|V\right(G\left)\right|-2}{\Delta \left(G\right)+1}⌋.\end{array}$$

Przybyło [10] proved that for all graphs $\mathrm{tvs}\left(G\right)<32\left|V\right(G\left)\right|/\delta \left(G\right)+8$ and that for r-regular graphs $\mathrm{tvs}\left(G\right)<8\left|V\right(G\left)\right|/r+3$. Anholcer et al. [11] improved this bound and proved that $\mathrm{tvs}\left(G\right)\le 3\left|V\right(G\left)\right|/\delta \left(G\right)+4$. Majerski and Przybyło [12] showed that $\mathrm{tvs}\left(G\right)\le \left(2+o\left(1\right)\right)\left|V\right(G\left)\right|/\delta \left(G\right)+4$, when $\delta \left(G\right)\ge \sqrt{\left|V\right(G\left)\right|}ln\left|V\left(G\right)\right|$.

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 $H_i$, $i=1,2,\dots ,t$. Then it is said that G admits an $(H_1,H_2,\dots ,H_t)$-(edge) covering. If every subgraph $H_i$ is isomorphic to a given graph H, then the graph G admits an H-covering.

Let G be a graph admitting an H-covering, and let $H_i$, $i=1,2,\dots ,t$, be all subgraphs of G isomorphic to H. For a subgraph $H\subseteq G$ under the total k-labeling $\psi$, the associated H-weight is defined as the sum of the labels of all vertices and edges belonging to H, i.e.,

$$w{t}_{\psi }\left(H\right)=\sum _{x\in V\left(H\right)}\psi \left(x\right)+\sum _{e\in E\left(H\right)}\psi \left(e\right).$$

A total k-labeling $\psi$ is called an H-irregular total k-labeling if the H-weights of all subgraphs isomorphic to H are distinct, that is, $w{t}_{\psi }\left(H_i\right)\ne w{t}_{\psi }\left(H_j\right)$ for every $1\le i<j\le t$. The minimum k for which G has an H-irregular total k-labeling is called the total H-irregularity strength of G, denoted by $\mathrm{ths}(G,H)$.

Clearly, the total $K_2$-irregularity strength of a graph G is equivalent to the total edge irregularity strength, that is $\mathrm{ths}(G,K_2)=\mathrm{tes}\left(G\right)$. The concept of the total H-irregularity strength and its lower bound were introduced by Ashraf et al. in [13]. The lower bound follows from the observation that the minimum possible H-weight is $\left|V\right(H\left)\right|+\left|E\right(H\left)\right|$, attained when every vertex and every edge of H receives the label 1. Since G contains t subgraphs isomorphic to H and all their weights must be distinct, the largest H-weight is at least $\left|V\right(H\left)\right|+\left|E\right(H\left)\right|+t-1$.

Theorem 1. 

[13]Let G be a graph admitting an H-covering containing exactly t subgraphs isomorphic to H. Then

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

(3)

Theorem 1 immediately implies the known lower bound for the total edge irregularity strength proved in [1].

Corollary 1. 

[13]Let $G=(V,E)$ be a graph having non-empty edge set. Then

$$\mathrm{ths}(G,K_2)=\mathrm{tes}\left(G\right)\ge ⌈\frac{\left|E\right(G\left)\right|+2}{3}⌉.$$

The lower bound (3) is sharp, as demonstrated by the following results for paths, fans and ladders. Recall that the fan graph $F_n$ is obtained by joining every vertex of the path $P_n$ to an additional vertex and $G\square H$ denotes the Cartesian product of graphs G and H.

Theorem 2. 

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

$$\mathrm{ths}(P_n,P_m)=⌈\frac{m+n-1}{2m-1}⌉.$$

Theorem 3. 

[14]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

$$\mathrm{ths}(F_n,F_m)=⌈\frac{2m+n}{3m}⌉.$$

Theorem 4. 

[14]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

$$\mathrm{ths}(L_n,L_m)=⌈\frac{4m+n-2}{5m-2}⌉.$$

In some cases the H-weights are not only distinct, but they also form a set of consecutive integers, allowing each copy of H to be uniquely identified by its weight. A natural relaxation of this requirement is to require only that the H-weights are distinct modulo the number t of subgraphs isomorphic to H. This motivates the following modular version of the total H-irregularity strength.

Let ${\mathbb{Z}}_t$ be the group of integers modulo t. A function $\psi :V\left(G\right)\cup E\left(G\right)\to \{1,2,\dots ,k\}$ is called a modular H-irregular total k-labeling if the weight function

$$\lambda :\{H_1,H_2,\dots ,H_t\}\to {\mathbb{Z}}_t$$

defined by

$$\lambda \left(H_i\right)\equiv w{t}_{\psi }\left(H_i\right)(modt)$$

is bijective, where $w{t}_{\psi }\left(H_i\right)$ denotes the H-weight of $H_i$, i.e.,

$$w{t}_{\psi }\left(H_i\right)=\sum _{x\in V\left(H_i\right)}\psi \left(x\right)+\sum _{e\in E\left(H_i\right)}\psi \left(e\right).$$

The value $\lambda \left(H_i\right)$ is called the modular weight of $H_i$. The modular total H-irregularity strength, denoted by $\mathrm{mths}(G,H)$, is the minimum k for which G admits a modular H-irregular total k-labeling.

Note that in [15], we introduced the vertex and edge versions of the modular H-irregularity strength, namely, the modular vertex H-irregularity strength $\mathrm{mvhs}(G,H)$ and the modular edge H-irregularity strength $\mathrm{mehs}(G,H)$, respectively. In that paper, we established the corresponding lower bounds and showed their sharpness. The modular total H-irregularity strength introduced here is a natural extension of both $\mathrm{mvhs}(G,H)$ and $\mathrm{mehs}(G,H)$, and its lower bound provides a unified extension of the corresponding lower bounds for these two invariants.

When $H\cong K_2$, the modular total H-irregularity strength $\mathrm{mths}(G,H)$ coincides with the modular total edge irregularity strength $\mathrm{mtes}\left(G\right)$, introduced by Bača et al. in [16]. The exact values of the modular total edge irregularity strength were determined in [16] for cycles, stars, n-sun graphs, friendship graphs, wheels, 3-regular circulant graphs $C_{2n}(1,n)$ and the two 4-regular circulant graphs $C_n(1,2)$ and $C_n(1,3)$.

In this paper, we study the existence of modular H-irregular total k-labelings for certain families of graphs and determine the exact values of the modular total H-irregularity strength for these graph families. For each graph family considered, we show that the lower bound is attained.

2. Main Results

Since every modular H-irregular total k-labeling of a graph G is also an H-irregular total k-labeling of G, we have

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

(4)

This immediately yields a lower bound for the corresponding modular graph invariant. In general, the converse of (4) does not hold. However, the next theorem provides a sufficient condition under which an H-irregular total k-labeling of a graph G is also a modular H-irregular total k-labeling.

Theorem 5. 

Let G be a simple graph admitting an H-covering with $\mathrm{ths}(G,H)=k$. If there exists an H-irregular total k-labeling whose H-weights form a set of consecutive integers, then

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

(5)

In the proof of Theorem 2, a $P_m$-irregular total $⌈(m+n-1)/(2m-1)⌉$-labeling of $P_n$ is constructed whose $P_m$-weights form a set of consecutive integers. Therefore, by Theorem 5 we obtain the next corollary.

Corollary 2. 

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

$$\mathrm{mths}(P_n,P_m)=⌈\frac{m+n-1}{2m-1}⌉.$$

The existence of an $F_m$-irregular total $⌈(2m+n)/3m⌉$-labeling of $F_n$ is established in the proof of Theorem 3. Moreover, it is shown that the difference between the weights of the subgraphs $F_m^{j+1}$ and $F_m^j$ is one for every $j=1,2,\dots ,n-m$. Hence, the corresponding $F_m$-weights form a set of consecutive integers, and Theorem 5 yields the following corollary.

Corollary 3. 

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

$$\mathrm{mths}(F_n,F_m)=⌈\frac{2m+n}{3m}⌉.$$

In the proof of Theorem 4, an $L_m$-irregular total labeling ${\psi }_m$ of $L_n$ is defined such that the $L_m$-weights under ${\psi }_m$ form a set of consecutive integers. Hence, by Theorem 5, ${\psi }_m$ is also a modular $L_m$-irregular total $⌈\left(4m+n-2\right)/\left(5m-2\right)⌉$-labeling of $L_n$, and thus we obtain the following result.

Corollary 4. 

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

$$\mathrm{mths}(L_n,L_m)=⌈\frac{4m+n-2}{5m-2}⌉.$$

The previous corollaries show that the lower bound (4) on the modular total H-irregularity strength is tight.

We next investigate the modular total H-irregularity strength for several graph families derived from ladder graphs.

The strong ladder $R_n\cong P_n⊠P_2$, $n\ge 2$, is the strong product of the paths $P_n$ and $P_2$. Its vertex set is $V\left(R_n\right)=\{x_i,y_i:i=1,2,\dots ,n\}$ and its edge set is $E\left(R_n\right)=\{x_i{x}_{i+1},y_i{y}_{i+1},x_i{y}_{i+1},x_{i+1}{y}_i:i=1,2,\dots ,n-1\}\cup \{x_i{y}_i:i=1,2,\dots ,n\}$. The following theorem gives the exact value of the modular total H-irregularity strength for the strong ladder $R_n$, where $H\cong R_m$, for $2\le m\le n$.

Theorem 6. 

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

$$\mathrm{mths}(R_n,R_m)=\mathrm{ths}(R_n,R_m)=⌈\frac{6m+n-4}{7m-4}⌉.$$

Proof. 

Let $n,m$ be positive integers, $2\le m\le n$. Put $k=⌈\left(6m+n-4\right)/\left(7m-4\right)⌉$.

The strong 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 $R_m$. From Theorem 1 it follows that $\mathrm{ths}(R_n,R_m)\ge k$. To prove the reverse inequality, we define a suitable $R_m$-irregular total labeling ${\psi }_m:V\left(R_n\right)\cup E\left(R_n\right)\to \{1,2,\dots ,k\}$ in the following way:

$$\begin{array}{cccc}\hfill {\psi }_m\left(x_i\right)& =⌈\frac{6m-4+i}{7m-4}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n,\hfill \\ \hfill {\psi }_m\left(y_i\right)& =⌈\frac{5m-4+i}{7m-4}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n,\hfill \\ \hfill {\psi }_m\left(x_i{y}_i\right)& =⌈\frac{2m-2+i}{7m-4}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n,\hfill \\ \hfill {\psi }_m\left(x_i{x}_{i+1}\right)& =⌈\frac{4m-3+i}{7m-4}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n-1,\hfill \\ \hfill {\psi }_m\left(y_i{y}_{i+1}\right)& =⌈\frac{3m-2+i}{7m-4}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n-1,\hfill \\ \hfill {\psi }_m\left(x_i{y}_{i+1}\right)& =⌈\frac{m-1+i}{7m-4}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n-1,\hfill \\ \hfill {\psi }_m\left(x_{i+1}{y}_i\right)& =⌈\frac{i}{7m-4}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n-1.\hfill \end{array}$$

For every $1\le i\le n-1$ we have

$${\psi }_m\left(x_{i+1}{y}_i\right)\le {\psi }_m\left(x_i{y}_{i+1}\right)\le {\psi }_m\left(x_i{y}_i\right)\le {\psi }_m\left(y_i{y}_{i+1}\right)\le {\psi }_m\left(x_i{x}_{i+1}\right)\le {\psi }_m\left(y_i\right)\le {\psi }_m\left(x_i\right)\le k,$$

thus ${\psi }_m$ is a total k-labeling. For the $R_m$-weight of the strong ladder $R_m^j$, $j=1,2,\dots ,n-m+1$, under the total k-labeling ${\psi }_m$ we obtain

$$\begin{array}{cc}\hfill w{t}_{{\psi }_m}\left(R_m^j\right)=& \sum _{x\in V\left(R_m^j\right)}{\psi }_m\left(x\right)+\sum _{e\in E\left(R_m^j\right)}{\psi }_m\left(e\right)={\displaystyle \sum _{i=j}^{m+j-1}}\left[{\psi }_m\left(x_i\right)+{\psi }_m\left(y_i\right)+{\psi }_m\left(x_i{y}_i\right)\right]\hfill \\ \hfill & +{\displaystyle \sum _{i=j}^{m+j-2}}\left[{\psi }_m\left(x_i{x}_{i+1}\right)+{\psi }_m\left(y_i{y}_{i+1}\right)+{\psi }_m\left(x_i{y}_{i+1}\right)+{\psi }_m\left(x_{i+1}{y}_i\right)\right].\hfill \end{array}$$

In particular, for $j=1$, we have

$$\begin{array}{cc}\hfill w{t}_{{\psi }_m}\left(R_m^1\right)=& {\displaystyle \sum _{i=1}^m}\left[{\psi }_m\left(x_i\right)+{\psi }_m\left(y_i\right)+{\psi }_m\left(x_i{y}_i\right)\right]\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^{m-1}}\left[{\psi }_m\left(x_i{x}_{i+1}\right)+{\psi }_m\left(y_i{y}_{i+1}\right)+{\psi }_m\left(x_i{y}_{i+1}\right)+{\psi }_m\left(x_{i+1}{y}_i\right)\right]\hfill \\ \hfill =& {\displaystyle \sum _{i=1}^m}\left[⌈\frac{6m-4+i}{7m-4}⌉+⌈\frac{5m-4+i}{7m-4}⌉+⌈\frac{2m-2+i}{7m-4}⌉\right]\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^{m-1}}\left[⌈\frac{4m-3+i}{7m-4}⌉+⌈\frac{3m-2+i}{7m-4}⌉+⌈\frac{m-1+i}{7m-4}⌉+⌈\frac{i}{7m-4}⌉\right]\hfill \\ \hfill =& {\displaystyle \sum _{i=1}^m}3+{\displaystyle \sum _{i=1}^{m-1}}4=3m+4(m-1)=7m-4.\hfill \end{array}$$

For the difference between the weights of the subgraphs $R_m^{j+1}$ and $R_m^j$ for $j=1,2,\dots ,n-m$ we obtain

$$\begin{array}{cc}\hfill w{t}_{{\psi }_m}\left(R_m^{j+1}\right)-w{t}_{{\psi }_m}\left(R_m^j\right)=& {\psi }_m\left(x_{m+j}\right)+{\psi }_m\left(y_{m+j}\right)+{\psi }_m\left(x_{m+j}{y}_{m+j}\right)+{\psi }_m\left(x_{m+j-1}{x}_{m+j}\right)\hfill \\ \hfill & +{\psi }_m\left(y_{m+j-1}{y}_{m+j}\right)+{\psi }_m\left(x_{m+j-1}{y}_{m+j}\right)+{\psi }_m\left(x_{m+j}{y}_{m+j-1}\right)\hfill \\ \hfill & -{\psi }_m\left(x_j\right)-{\psi }_m\left(y_j\right)-{\psi }_m\left(x_j{y}_j\right)-{\psi }_m\left(x_j{x}_{j+1}\right)-{\psi }_m\left(y_j{y}_{j+1}\right)\hfill \\ \hfill & -{\psi }_m\left(x_j{y}_{j+1}\right)-{\psi }_m\left(x_{j+1}{y}_j\right)\hfill \\ \hfill =& ⌈\frac{6m-4+m+j}{7m-4}⌉+⌈\frac{5m-4+m+j}{7m-4}⌉+⌈\frac{2m-2+m+j}{7m-4}⌉+⌈\frac{4m-3+m+j-1}{7m-4}⌉\hfill \\ \hfill & +⌈\frac{3m-2+m+j-1}{7m-4}⌉+⌈\frac{m-1+m+j-1}{7m-4}⌉+⌈\frac{m+j-1}{7m-4}⌉-⌈\frac{6m-4+j}{7m-4}⌉\hfill \\ \hfill & -⌈\frac{5m-4+j}{7m-4}⌉-⌈\frac{2m-2+j}{7m-4}⌉-⌈\frac{4m-3+j}{7m-4}⌉-⌈\frac{3m-2+j}{7m-4}⌉\hfill \\ \hfill & -⌈\frac{m-1+j}{7m-4}⌉-⌈\frac{j}{7m-4}⌉=1.\hfill \end{array}$$

Hence, the $R_m$-weights form a sequence of consecutive integers from $7m-4$ to $6m+n-4$. Therefore, by Theorem 5, we conclude that $\mathrm{mths}(R_n,R_m)=\mathrm{ths}(R_n,R_m)=k$. This completes the proof. □

Recall that $L_n\cong P_n\square P_2$, $n\ge 2$, denotes the ladder graph. The ladder-wing graph$L{W}_n$, $n\ge 2$, is obtained from $L_n$ by joining every vertex of $L_n$ to an additional vertex z. Thus, $V\left(L{W}_n\right)=\{z,x_i,y_i:i=1,2,\dots ,n\}$ and $E\left(L{W}_n\right)=\{x_i{x}_{i+1},y_i{y}_{i+1}:i=1,2,\dots ,n-1\}\cup \{x_i{y}_i,z{x}_i,z{y}_i:i=1,2,\dots ,n\}$. The next theorem gives the exact value of the modular total H-irregularity strength for $L{W}_n$, where $H\cong L{W}_m$, $2\le m\le n$.

Theorem 7. 

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

$$\mathrm{mths}(L{W}_n,L{W}_m)=\mathrm{ths}(L{W}_n,L{W}_m)=⌈\frac{6m+n-2}{7m-2}⌉.$$

Proof. 

Let $L{W}_n$, $n\ge 2$, be a ladder-wing graph. For every m, $2\le m\le n$, the ladder-wing graph $L{W}_n$ admits a $L{W}_m$-covering with exactly $t=n-m+1$ subgraphs. By Theorem 1 we have $\mathrm{ths}(L{W}_n,L{W}_m)\ge ⌈\left(6m+n-1\right)/\left(7m-1\right)⌉$. On the other hand, let $\psi$ be an arbitrary total k-labeling of $L{W}_n$. The smallest $L{W}_m$-weight is at least $7m-2+\psi \left(z\right)$. Since there are exactly $t=n-m+1$ subgraphs isomorphic to $L{W}_m$, the largest $L{W}_m$-weight is at least $6m+n-2+\psi \left(z\right)$ and at most $(7m-2)k+\psi \left(z\right)$. Hence $6m+n-2+\psi \left(z\right)\le (7m-2)k+\psi \left(z\right)$ which implies $k\ge ⌈\left(6m+n-2\right)/\left(7m-2\right)⌉$. To prove the reverse inequality, it suffices to construct an $L{W}_m$-irregular total $⌈\left(6m+n-2\right)/\left(7m-2\right)⌉$-labeling.

Let ${\psi }_m:V\left(L{W}_n\right)\cup E\left(L{W}_n\right)\to \{1,2,\dots ,k\}$ be a total labeling defined such that

$$\begin{array}{cc}\hfill {\psi }_m\left(z\right)& =1,\hfill \\ \hfill {\psi }_m\left(x_i\right)& =⌈\frac{5m-2+i}{7m-2}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n,\hfill \\ \hfill {\psi }_m\left(y_i\right)& =⌈\frac{6m-2+i}{7m-2}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n,\hfill \\ \hfill {\psi }_m\left(x_i{y}_i\right)& =⌈\frac{2m+i}{7m-2}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n,\hfill \\ \hfill {\psi }_m\left(x_i{x}_{i+1}\right)& =⌈\frac{3m+i}{7m-2}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n-1,\hfill \\ \hfill {\psi }_m\left(y_i{y}_{i+1}\right)& =⌈\frac{4m-1+i}{7m-2}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n-1,\hfill \\ \hfill {\psi }_m\left(x_{i}z\right)& =⌈\frac{i}{7m-2}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n,\hfill \\ \hfill {\psi }_m\left(y_{i}z\right)& =⌈\frac{m+i}{7m-2}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n.\hfill \end{array}$$

For every admissible index i, we have

$${\psi }_m\left(x_{i}z\right)\le {\psi }_m\left(y_{i}z\right)\le {\psi }_m\left(x_i{y}_i\right)\le {\psi }_m\left(x_i{x}_{i+1}\right)\le {\psi }_m\left(y_i{y}_{i+1}\right)\le {\psi }_m\left(x_i\right)\le {\psi }_m\left(y_i\right)\le ⌈\frac{6m+n-2}{7m-2}⌉.$$

Thus, for $j=1,2,\dots ,n-m+1$, we have

$$\begin{array}{cc}\hfill w{t}_{{\psi }_m}\left(L{W}_m^j\right)=& \sum _{x\in V\left(L{W}_m^j\right)}{\psi }_m\left(x\right)+\sum _{e\in E\left(L{W}_m^j\right)}{\psi }_m\left(e\right)={\displaystyle \sum _{i=j}^{m+j-1}}\left[{\psi }_m\left(x_i\right)+{\psi }_m\left(y_i\right)+{\psi }_m\left(x_i{y}_i\right)\right.\hfill \\ \hfill & +\left.{\psi }_m\left(x_{i}z\right)+{\psi }_m\left(y_{i}z\right)\right]+{\psi }_m\left(z\right)+{\displaystyle \sum _{i=j}^{m+j-2}}\left[{\psi }_m\left(x_i{x}_{i+1}\right)+{\psi }_m\left(y_i{y}_{i+1}\right)\right].\hfill \end{array}$$

For $j=1$, we obtain

$$\begin{array}{cc}\hfill w{t}_{{\psi }_m}\left(L{W}_m^1\right)=& {\displaystyle \sum _{i=1}^m}\left[{\psi }_m\left(x_i\right)+{\psi }_m\left(y_i\right)+{\psi }_m\left(x_i{y}_i\right)+{\psi }_m\left(x_{i}z\right)+{\psi }_m\left(y_{i}z\right)\right]+{\psi }_m\left(z\right)\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^{m-1}}\left[{\psi }_m\left(x_i{x}_{i+1}\right)+{\psi }_m\left(y_i{y}_{i+1}\right)\right]\hfill \\ \hfill =& {\displaystyle \sum _{i=1}^m}\left[⌈\frac{5m-2+i}{7m-2}⌉+⌈\frac{6m-2+i}{7m-2}⌉+⌈\frac{2m+i}{7m-2}⌉+⌈\frac{i}{7m-2}⌉+⌈\frac{m+i}{7m-2}⌉\right]+1\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^{m-1}}\left[⌈\frac{3m+i}{7m-2}⌉+⌈\frac{4m-1+i}{7m-2}⌉\right]={\displaystyle \sum _{i=1}^m}5+1+{\displaystyle \sum _{i=1}^{m-1}}2=5m+1+2(m-1)\hfill \\ \hfill =& 7m-1.\hfill \end{array}$$

Now consider the difference between the weights of the subgraphs $L{W}_m^{j+1}$ and $L{W}_m^j$, where $j=1,2,\dots ,n-m$.

$$\begin{array}{cc}\hfill w{t}_{{\psi }_m}\left(L{W}_m^{j+1}\right)-w{t}_{{\psi }_m}\left(L{W}_m^j\right)=& {\psi }_m\left(x_{m+j}\right)+{\psi }_m\left(y_{m+j}\right)+{\psi }_m\left(x_{m+j-1}{x}_{m+j}\right)+{\psi }_m\left(y_{m+j-1}{y}_{m+j}\right)\hfill \\ \hfill & +{\psi }_m\left(x_{m+j}{y}_{m+j}\right)+{\psi }_m\left(x_{m+j}z\right)+{\psi }_m\left(y_{m+j}z\right)-{\psi }_m\left(x_j\right)\hfill \\ \hfill & -{\psi }_m\left(y_j\right)-{\psi }_m\left(x_j{x}_{j+1}\right)-{\psi }_m\left(y_j{y}_{j+1}\right)-{\psi }_m\left(x_j{y}_j\right)-{\psi }_m\left(x_{j}z\right)\hfill \\ \hfill & -{\psi }_m\left(y_{j}z\right)=⌈\frac{6m-2+m+j}{7m-2}⌉+⌈\frac{5m-2+m+j}{7m-2}⌉+⌈\frac{4m-1+m+j-1}{7m-2}⌉\hfill \\ \hfill & +⌈\frac{3m+m+j-1}{7m-2}⌉+⌈\frac{2m+m+j}{7m-2}⌉+⌈\frac{m+m+j}{7m-2}⌉+⌈\frac{m+j}{7m-2}⌉\hfill \\ \hfill & -⌈\frac{6m-2+j}{7m-2}⌉-⌈\frac{5m-2+j}{7m-2}⌉-⌈\frac{4m-1+j}{7m-2}⌉-⌈\frac{3m+j}{7m-2}⌉-⌈\frac{2m+j}{7m-2}⌉\hfill \\ \hfill & -⌈\frac{m+j}{7m-2}⌉-⌈\frac{j}{7m-2}⌉=1.\hfill \end{array}$$

Hence, the $L{W}_m^j$-weights form the sequence of consecutive integers from $7m-1$ up to $6m+n-1$. Therefore, by Theorem 5, we conclude that $\mathrm{mths}(L{W}_n,L{W}_m)=\mathrm{ths}(L{W}_n,L{W}_m)=⌈\left(6m+n-2\right)/\left(7m-2\right)⌉$. □

The strong ladder-wing graph $R{W}_n$, $n\ge 2$, is obtained from the strong ladder $R_n\cong P_n⊠P_2$ by joining every vertex of $R_n$ to an additional vertex z. Thus, $V\left(R{W}_n\right)=\{z,x_i,y_i:i=1,2,\dots ,n\}$ and $E\left(R{W}_n\right)=\{x_i{x}_{i+1},y_i{y}_{i+1},x_i{y}_{i+1},x_{i+1}{y}_i:i=1,2,\dots ,n-1\}\cup \{x_i{y}_i,x_{i}z,y_{i}z:i=1,2,\dots ,n\}$.

The following theorem gives the exact value of the modular total H-irregularity strength for the strong ladder-wing graph $R{W}_n$, where $H\cong R{W}_m$, $2\le m\le n$.

Theorem 8. 

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

$$\mathrm{mths}(R{W}_n,R{W}_m)=\mathrm{ths}(R{W}_n,R{W}_m)=⌈\frac{8m+n-4}{9m-4}⌉.$$

Proof. 

Let n and m be positive integers, $2\le m\le n$. The strong ladder-wing graph $R{W}_n$ admits an $R{W}_m$-covering with exactly $t=n-m+1$ subgraphs isomorphic to $R{W}_m$. Hence, by Theorem 1 it follows that $\mathrm{ths}(R{W}_n,R{W}_m)\ge ⌈\left(8m+n-3\right)/\left(9m-3\right)⌉$. On the other hand, let $\psi$ be an arbitrary $R{W}_m$-irregular total k-labeling of $R{W}_n$. The smallest $R{W}_m$-weight is at least $9m-4+\psi \left(z\right)$. Since there are exactly $t=n-m+1$ subgraphs isomorphic to $R{W}_m$, the largest $R{W}_m$-weight is at least $8m+n-4+\psi \left(z\right)$ and at most $(9m-4)k+\psi \left(z\right)$. Therefore, $8m+n-4+\psi \left(z\right)\le (9m-4)k+\psi \left(z\right)$, which implies $k\ge ⌈\left(8m+n-4\right)/\left(9m-4\right)⌉$. It remains to construct an $R{W}_m$-irregular total $⌈\left(8m+n-4\right)/\left(9m-4\right)⌉$-labeling of $R{W}_n$.

Let ${\psi }_m:V\left(R{W}_n\right)\cup E\left(R{W}_n\right)\to \{1,2,\dots ,k\}$ be a total labeling defined as follows:

$$\begin{array}{cc}\hfill {\psi }_m\left(z\right)& =1,\hfill \\ \hfill {\psi }_m\left(x_i\right)& =⌈\frac{8m-4+i}{9m-4}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n,\hfill \\ \hfill {\psi }_m\left(y_i\right)& =⌈\frac{7m-4+i}{9m-4}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n,\hfill \\ \hfill {\psi }_m\left(x_i{y}_i\right)& =⌈\frac{4m-2+i}{9m-4}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n,\hfill \\ \hfill {\psi }_m\left(x_i{x}_{i+1}\right)& =⌈\frac{6m-3+i}{9m-4}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n-1,\hfill \\ \hfill {\psi }_m\left(y_i{y}_{i+1}\right)& =⌈\frac{5m-2+i}{9m-4}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n-1,\hfill \\ \hfill {\psi }_m\left(x_i{y}_{i+1}\right)& =⌈\frac{3m-1+i}{9m-4}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n-1,\hfill \\ \hfill {\psi }_m\left(x_{i+1}{y}_i\right)& =⌈\frac{2m+i}{9m-4}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n-1,\hfill \\ \hfill {\psi }_m\left(x_{i}z\right)& =⌈\frac{m+i}{9m-4}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n,\hfill \\ \hfill {\psi }_m\left(y_{i}z\right)& =⌈\frac{i}{9m-4}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n.\hfill \end{array}$$

For every admissible index i,

$$\begin{array}{cc}\hfill {\psi }_m\left(y_{i}z\right)\le & {\psi }_m\left(x_{i}z\right)\le {\psi }_m\left(x_{i+1}{y}_i\right)\le {\psi }_m\left(x_i{y}_{i+1}\right)\le {\psi }_m\left(x_i{y}_i\right)\le {\psi }_m\left(y_i{y}_{i+1}\right)\le {\psi }_m\left(x_i{x}_{i+1}\right)\hfill \\ \hfill \le & {\psi }_m\left(y_i\right)\le {\psi }_m\left(x_i\right)\le ⌈\frac{8m+n-4}{9m-4}⌉.\hfill \end{array}$$

Thus, ${\psi }_m$ is a total $⌈\left(8m+n-4\right)/\left(9m-4\right)⌉$-labeling.

For the $R{W}_m$-weight of the strong ladder-wing graph $R{W}_m^j$, $j=1,2,\dots ,n-m+1$, under the total labeling ${\psi }_m$, we obtain

$$\begin{array}{cc}\hfill w{t}_{{\psi }_m}\left(R{W}_m^j\right)=& \sum _{x\in V\left(R{W}_m^j\right)}{\psi }_m\left(x\right)+\sum _{e\in E\left(R{W}_m^j\right)}{\psi }_m\left(e\right)\hfill \\ \hfill =& {\displaystyle \sum _{i=j}^{m+j-1}}\left[{\psi }_m\left(x_i\right)+{\psi }_m\left(y_i\right)+{\psi }_m\left(x_i{y}_i\right)+{\psi }_m\left(x_{i}z\right)+{\psi }_m\left(y_{i}z\right)\right]+{\psi }_m\left(z\right)\hfill \\ & +{\displaystyle \sum _{i=j}^{m+j-2}}\left[{\psi }_m\left(x_i{x}_{i+1}\right)+{\psi }_m\left(y_i{y}_{i+1}\right)+{\psi }_m\left(x_i{y}_{i+1}\right)+{\psi }_m\left(x_{i+1}{y}_i\right)\right].\hfill \end{array}$$

Substituting $j=1$ into the above expression and simplifying, we obtain

$$w{t}_{{\psi }_m}\left(R{W}_m^1\right)=9m-3.$$

Next, we evaluate the difference between the weights of the subgraphs $R{W}_m^{j+1}$ and $R{W}_m^j$ for $j=1,2,\dots ,n-m$.

$$\begin{array}{cc}\hfill w{t}_{{\psi }_m}\left(R{W}_m^{j+1}\right)-w{t}_{{\psi }_m}\left(R{W}_m^j\right)=& {\psi }_m\left(x_{m+j}\right)+{\psi }_m\left(y_{m+j}\right)+{\psi }_m\left(x_{m+j}{y}_{m+j}\right)+{\psi }_m\left(x_{m+j-1}{x}_{m+j}\right)\hfill \\ \hfill & +{\psi }_m\left(y_{m+j-1}{y}_{m+j}\right)+{\psi }_m\left(x_{m+j-1}{y}_{m+j}\right)+{\psi }_m\left(x_{m+j}{y}_{m+j-1}\right)\hfill \\ \hfill & +{\psi }_m\left(x_{m+j}z\right)+{\psi }_m\left(y_{m+j}z\right)-{\psi }_m\left(x_j\right)-{\psi }_m\left(y_j\right)-{\psi }_m\left(x_j{y}_j\right)\hfill \\ \hfill & -{\psi }_m\left(x_j{x}_{j+1}\right)-{\psi }_m\left(y_j{y}_{j+1}\right)-{\psi }_m\left(x_j{y}_{j+1}\right)-{\psi }_m\left(x_{j+1}{y}_j\right)\hfill \\ \hfill & -{\psi }_m\left(x_{j}z\right)-{\psi }_m\left(y_{j}z\right)=⌈\frac{8m-4+m+j}{9m-4}⌉+⌈\frac{7m-4+m+j}{9m-4}⌉\hfill \\ \hfill & +⌈\frac{4m-2+m+j}{9m-4}⌉+⌈\frac{6m-3+m+j-1}{9m-4}⌉+⌈\frac{5m-2+m+j-1}{9m-4}⌉\hfill \\ \hfill & +⌈\frac{3m-1+m+j-1}{9m-4}⌉+⌈\frac{2m+m+j-1}{9m-4}⌉+⌈\frac{m+m+j}{9m-4}⌉+⌈\frac{m+j}{9m-4}⌉\hfill \\ \hfill & -⌈\frac{8m-4+j}{9m-4}⌉-⌈\frac{7m-4+j}{9m-4}⌉-⌈\frac{4m-2+j}{9m-4}⌉-⌈\frac{6m-3+j}{9m-4}⌉\hfill \\ \hfill & -⌈\frac{5m-2+j}{9m-4}⌉-⌈\frac{3m-1+j}{9m-4}⌉-⌈\frac{2m+j}{9m-4}⌉-⌈\frac{m+j}{9m-4}⌉-⌈\frac{j}{9m-4}⌉=1.\hfill \end{array}$$

Since the $R{W}_m$-weights form a sequence of consecutive integers from $9m-3$ to $8m+n-3$, Theorem 5 implies that

$$\mathrm{mths}(R{W}_n,R{W}_m)=\mathrm{ths}(R{W}_n,R{W}_m)=⌈\frac{8m+n-4}{9m-4}⌉.$$

□

3. Concluding Remarks

In this paper, we introduced the modular total H-irregularity strength as a modular counterpart of the total H-irregularity strength of graphs. We established a sufficient condition under which these two graph invariants coincide. As applications of this result, we determined the exact values of the modular total H-irregularity strength for paths, fans, ladders, strong ladders, ladder-wing graphs, and strong ladder-wing graphs. In all these cases, the obtained values attain the corresponding lower bound, showing that this bound is sharp.

The friendship graph provides an interesting example illustrating the difference between the classical and modular settings. Recall that a friendship graph $f_n$, $n\ge 1$, consists of n triangles sharing a common central vertex. Consequently, the number of subgraphs of $f_n$ isomorphic to $f_m$ equals $\left({\displaystyle \genfrac{}{}{0pt}{}{n}{m}}\right)$, making the determination of the modular total H-irregularity strength considerably more difficult than for the graph families studied in this paper. For the classical total H-irregularity strength, the existence of many copies of H does not necessarily prevent the existence of suitable labelings, and it is natural to expect that techniques based on special integer sequences, successfully used in related graph labeling problems, could also be adapted to this setting.

For the modular variant, however, the requirement that the H-weights represent all residue classes modulo the number of copies makes the problem substantially more restrictive. Nevertheless, one may instead consider a fixed covering consisting of copies induced by consecutive triangles. We therefore consider the following restricted version of the problem, in which irregularity is required only with respect to the members of the fixed covering.

For $j=1,2,\dots ,n-m+1$, let $f_m^j$ denote the subgraph of $f_n$ induced by the central vertex and the vertices of the m consecutive triangles with indices $j,j+1,\dots ,j+m-1$. These subgraphs form an $f_m$-covering of $f_n$.

Theorem 9. 

Let $f_n$, $n\ge 1$, be a friendship graph, and let m be a positive integer such that $1\le m\le n$. With respect to the $f_m$-covering $\{f_m^1,f_m^2,\dots ,f_m^{n-m+1}\}$ defined above, the corresponding total and modular total $f_m$-irregularity strengths are equal and have value

$$⌈\frac{4m+n}{5m}⌉.$$

Proof. 

Let the friendship graph $f_n$ have vertex set $V\left(f_n\right)=\{z,x_i,y_i:i=1,2,\dots ,n\}$ and edge set $E\left(f_n\right)=\{x_i{y}_i,x_{i}z,y_{i}z:i=1,2,\dots ,n\}$. Let m be a positive integer such that $1\le m\le n$.

Consider the fixed $f_m$-covering $\mathcal{C}=\{f_m^1,f_m^2,\dots ,f_m^{n-m+1}\}$, where, for $j=1,2,\dots ,n-m+1$, $V\left(f_m^j\right)=\{z,x_i,y_i:i=j,j+1,\dots ,j+m-1\}$. Thus, $\mathcal{C}$ consists of $t=n-m+1$ subgraphs isomorphic to $f_m$. We consider irregularity only with respect to the members of $\mathcal{C}$.

Let $\psi$ be an arbitrary $f_m$-irregular total k-labeling with respect to $\mathcal{C}$. The smallest $f_m$-weight is at least $5m+\psi \left(z\right)$. Since $\mathcal{C}$ contains $t=n-m+1$ subgraphs and their weights must be distinct, the largest $f_m$-weight is at least $4m+n+\psi \left(z\right)$ and at most $5mk+\psi \left(z\right)$. Hence, $4m+n+\psi \left(z\right)\le 5mk+\psi \left(z\right)$. Thus $k\ge ⌈\left(4m+n\right)/5m⌉$.

To attain this bound, define a labeling ${\psi }_m$ of vertices and edges of $f_n$ in the following way:

$$\begin{array}{cc}\hfill {\psi }_m\left(z\right)& =1,\hfill \\ \hfill {\psi }_m\left(x_i\right)& =⌈\frac{2m+i}{5m}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n,\hfill \\ \hfill {\psi }_m\left(y_i\right)& =⌈\frac{m+i}{5m}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n,\hfill \\ \hfill {\psi }_m\left(x_i{y}_i\right)& =⌈\frac{3m+i}{5m}⌉,\hfill & \hfill & \mathrm{for}i=1,2,\dots ,n,\hfill \\ \hfill {\psi }_m\left(x_{i}z\right)& =⌈\frac{i}{5m}⌉,\hfill & & \mathrm{for}i=1,2,\dots ,n,\hfill \\ \hfill {\psi }_m\left(y_{i}z\right)& =⌈\frac{4m+i}{5m}⌉,\hfill & & \mathrm{for}i=1,2,\dots ,n.\hfill \end{array}$$

For every $1\le i\le n$ we have

$${\psi }_m\left(x_{i}z\right)\le {\psi }_m\left(y_i\right)\le {\psi }_m\left(x_i\right)\le {\psi }_m\left(x_i{y}_i\right)\le {\psi }_m\left(y_{i}z\right)\le ⌈\frac{4m+n}{5m}⌉.$$

Thus, ${\psi }_m$ is a total $⌈\left(4m+n\right)/5m⌉$-labeling.

For $j=1,2,\dots ,n-m+1$, the $f_m$-weight of $f_m^j$ under ${\psi }_m$ is

$$\begin{array}{cc}\hfill w{t}_{{\psi }_m}\left(f_m^j\right)=& \sum _{x\in V\left(f_m^j\right)}{\psi }_m\left(x\right)+\sum _{e\in E\left(f_m^j\right)}{\psi }_m\left(e\right)={\displaystyle \sum _{i=j}^{m+j-1}}\left[{\psi }_m\left(x_i\right)+{\psi }_m\left(y_i\right)\right]+{\psi }_m\left(z\right)\hfill \\ \hfill & +{\displaystyle \sum _{i=j}^{m+j-1}}\left[{\psi }_m\left(x_i{y}_i\right)+{\psi }_m\left(x_{i}z\right)+{\psi }_m\left(y_{i}z\right)\right].\hfill \end{array}$$

In particular, for $j=1$, we have

$$\begin{array}{cc}\hfill w{t}_{{\psi }_m}\left(f_m^1\right)=& {\displaystyle \sum _{i=1}^m}\left[{\psi }_m\left(x_i\right)+{\psi }_m\left(y_i\right)\right]+{\psi }_m\left(z\right)+{\displaystyle \sum _{i=1}^m}\left[{\psi }_m\left(x_i{y}_i\right)+{\psi }_m\left(x_{i}z\right)+{\psi }_m\left(y_{i}z\right)\right]\hfill \\ \hfill =& {\displaystyle \sum _{i=1}^m}\left[⌈\frac{2m+i}{5m}⌉+⌈\frac{m+i}{5m}⌉\right]+1+{\displaystyle \sum _{i=1}^m}\left[⌈\frac{3m+i}{5m}⌉+⌈\frac{i}{5m}⌉+⌈\frac{4m+i}{5m}⌉\right]\hfill \\ \hfill =& {\displaystyle \sum _{i=1}^m}2+1+{\displaystyle \sum _{i=1}^m}3=2m+1+3m=5m+1.\hfill \end{array}$$

For $j=1,2,\dots ,n-m$, consider the difference between the weights of $f_m^{j+1}$ and $f_m^j$. We obtain

$$\begin{array}{cc}\hfill w{t}_{{\psi }_m}\left(f_m^{j+1}\right)-w{t}_{{\psi }_m}\left(f_m^j\right)=& {\psi }_m\left(x_{m+j}\right)+{\psi }_m\left(y_{m+j}\right)+{\psi }_m\left(x_{m+j}{y}_{m+j}\right)+{\psi }_m\left(x_{m+j}z\right)\hfill \\ \hfill & +{\psi }_m\left(y_{m+j}z\right)-{\psi }_m\left(x_j\right)-{\psi }_m\left(y_j\right)-{\psi }_m\left(x_j{y}_j\right)-{\psi }_m\left(x_{j}z\right)-{\psi }_m\left(y_{j}z\right)\hfill \\ \hfill =& ⌈\frac{2m+m+j}{5m}⌉+⌈\frac{m+m+j}{5m}⌉+⌈\frac{3m+m+j}{5m}⌉+⌈\frac{m+j}{5m}⌉+⌈\frac{4m+m+j}{5m}⌉\hfill \\ \hfill & -⌈\frac{2m+j}{5m}⌉-⌈\frac{m+j}{5m}⌉-⌈\frac{3m+j}{5m}⌉-⌈\frac{j}{5m}⌉-⌈\frac{4m+j}{5m}⌉=1.\hfill \end{array}$$

Hence, the $f_m$-weights of the members of $\mathcal{C}$ form a sequence of consecutive integers from $5m+1$ to $4m+n+1$. In particular, they are pairwise distinct and also represent all residue classes modulo $n-m+1$. Therefore, ${\psi }_m$ is simultaneously an $f_m$-irregular total labeling and a modular $f_m$-irregular total labeling with respect to the covering $\mathcal{C}$, using labels at most $⌈\left(4m+n\right)/5m⌉$. Together with the lower bound above, this proves the claim. □

The above result suggests that considering a fixed H-covering may be particularly useful for graph families in which the number of subgraphs isomorphic to H is large. Further investigation of this restricted setting, as well as the use of suitable integer sequences for the unrestricted problem, may provide a way to determine the corresponding irregularity strengths for broader classes of graphs.