Modular Total H-Irregularity Strength of Graphs

Abstract

We investigate the modular total H-irregularity strength, a modular variant of the total H-irregularity strength of graphs. We establish a sufficient condition under which these two graph invariants coincide. As applications of this result, we determine the exact values of the modular total H-irregularity strength for several graph families. In each case, the obtained value attains the corresponding lower bound, proving that the bound is sharp.

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 $⌈(\Delta (G)+1)/2⌉$. This conjecture has been verified for complete graphs and complete bipartite graphs in [3,4], for large dense graphs with $\left(\right|E\left(G\right)|+2)/3\le (\Delta (G)+1)/2$ in [5], for generalized Petersen graphs in [6], for generalized prisms in [7], for the categorical product of two cycles in [8] and for the corona product of a path with certain graphs in [9].

The vertex analog of this invariant, the total vertex irregularity strength, have been extensively studied; see, for example, [10,11,12].

These classical invariants distinguish individual edges or vertices by their weights. The total H-irregularity strength extends this approach by requiring distinct weights for subgraphs isomorphic to a prescribed graph H. Although the total H-irregularity strength and the modular vertex and edge H-irregularity strengths have been investigated, their modular total counterpart has not yet been studied. This is the research gap addressed in the present paper.

An edge-covering of G is a family $\mathcal{C}=\{H_1,H_2,\dots ,H_t\}$ of subgraphs of G such that every edge of G belongs to at least one member of $\mathcal{C}$. If every $H_i$ is isomorphic to a given graph H, then $\mathcal{C}$ is called an H-covering of G, see [13].

Let G be a graph admitting an H-covering, and let $H_1,H_2,\dots ,H_t$ denote 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 $\mathrm{ths}(G,H)$.

The total H-irregularity strength is finite for every graph admitting an H-covering. More precisely, it was proved in [14] that $\mathrm{ths}(G,H)\le 2^{\left|E\right(G\left)\right|-1}$.

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 [14]. 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

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

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

(2)

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

Corollary 1

([14]). Let $G=(V,E)$ be a graph having a 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 (2) 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

([14]). 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

([15]). 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

([15]). 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 $H_1,H_2,\dots ,H_t$ be all subgraphs of G isomorphic to H, and 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

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

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. If no modular H-irregular total labeling of G exists, we set $\mathrm{mths}(G,H)=\infty$.

Note that in [16], 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 [17]. The exact values of the modular total edge irregularity strength were determined in [17] 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. We also establish a general necessary condition for the existence of such labelings and identify infinite classes for which the modular parameter is infinite. For paths, fans, ladders, strong ladders, ladder-wing graphs, and strong ladder-wing graphs, we construct modular H-irregular total labelings attaining the corresponding lower bounds.

2. Main Results

We first give a necessary condition for the existence of a modular H-irregular total labeling. For each $x\in V\left(G\right)\cup E\left(G\right)$, let $c\left(x\right)$ denote the number of subgraphs of G isomorphic to H that contain x.

Theorem 5.

Let G be a graph containing exactly t subgraphs isomorphic to H, and let d be the greatest common divisor of t and all the numbers $c\left(x\right)$, $x\in V\left(G\right)\cup E\left(G\right)$. If G admits a modular H-irregular total labeling, then d must divide $t(t-1)/2$, that is,

$$d\mid \frac{t(t-1)}{2}.$$

Proof. 

Let $H_1,H_2,\dots ,H_t$ be all subgraphs of G isomorphic to H, and suppose that $\psi$ is a modular H-irregular total labeling of G. Since the H-weights represent all residue classes modulo t, we have

$${\displaystyle \sum _{i=1}^t}w{t}_{\psi }\left(H_i\right)\equiv {\displaystyle \sum _{j=0}^{t-1}}j=\frac{t(t-1)}{2}\left(\mathrm{mod}t\right).$$

On the other hand, each label $\psi \left(x\right)$, where $x\in V\left(G\right)\cup E\left(G\right)$, occurs in this sum exactly $c\left(x\right)$ times. Therefore,

$${\displaystyle \sum _{i=1}^t}w{t}_{\psi }\left(H_i\right)=\sum _{x\in V\left(G\right)\cup E\left(G\right)}c\left(x\right)\psi \left(x\right).$$

By the definition of d, every $c\left(x\right)$ and also t are divisible by d. Hence,

$${\displaystyle \sum _{i=1}^t}w{t}_{\psi }\left(H_i\right)\equiv 0\left(\mathrm{mod}d\right).$$

Since congruence modulo t implies congruence modulo d, it follows that

$$\frac{t(t-1)}{2}\equiv 0\left(\mathrm{mod}d\right).$$

Therefore, $d\mid t(t-1)/2$. □

Corollary 2.

Let G be a graph containing exactly t subgraphs isomorphic to H. If $t\equiv 2\left(\mathrm{mod}4\right)$ and $c\left(x\right)$ is even for every $x\in V\left(G\right)\cup E\left(G\right)$, then G admits no modular H-irregular total labeling.

Proof. 

Under the assumptions, d is even, whereas $t(t-1)/2$ is odd. Therefore, d does not divide $t(t-1)/2$, and the result follows from Theorem 5. □

It is known that $\mathrm{ths}(G,H)$ is finite for every graph admitting an H-covering, see [14]. In contrast, Corollary 2 shows that $\mathrm{mths}(G,H)$ need not be finite.

Corollary 3.

Let m and n be positive integers such that $2\le m\le n$. If $\left({\displaystyle \genfrac{}{}{0pt}{}{n}{m}}\right)\equiv 2\left(\mathrm{mod}4\right)$ and both $\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{m-1}}\right)$ and $\left({\displaystyle \genfrac{}{}{0pt}{}{n-2}{m-2}}\right)$ are even, then

$$\mathrm{mths}(K_n,K_m)=\infty .$$

Proof. 

The copies of $K_m$ in $K_n$ are determined by choosing m vertices from the n vertices of $K_n$. Hence, $K_n$ contains $\left({\displaystyle \genfrac{}{}{0pt}{}{n}{m}}\right)$ subgraphs isomorphic to $K_m$. A fixed vertex of $K_n$ belongs to $\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{m-1}}\right)$ copies of $K_m$, since the remaining $m-1$ vertices can be chosen from the other $n-1$ vertices. Similarly, a fixed edge belongs to $\left({\displaystyle \genfrac{}{}{0pt}{}{n-2}{m-2}}\right)$ copies of $K_m$. The result follows from Corollary 2. □

The general upper bound for the total H-irregularity strength established in [14] implies that $\mathrm{ths}(K_n,K_m)<\infty$. Hence, under the assumptions of Corollary 3, a $K_m$-irregular total labeling of $K_n$ exists, whereas a modular $K_m$-irregular total labeling does not exist.

As a particular case, let $m=4$. Direct calculation shows that the assumptions of Corollary 3 are satisfied when $n\equiv 10$ or $11\left(\mathrm{mod}16\right)$. Consequently, for every nonnegative integer q,

$$\mathrm{ths}(K_{16q+10},K_4)<\infty ,\mathrm{mths}(K_{16q+10},K_4)=\infty ,$$

and

$$\mathrm{ths}(K_{16q+11},K_4)<\infty ,\mathrm{mths}(K_{16q+11},K_4)=\infty .$$

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

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

(3)

This immediately yields a lower bound for the corresponding modular graph invariant. In general, the converse of (3) 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 6.

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

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

(4)

Proof. 

Let $H_1,H_2,\dots ,H_t$ be all subgraphs of G isomorphic to H, and let $\psi$ be an H-irregular total k-labeling whose H-weights form a set of consecutive integers. Since there are exactly t such weights, they can be written as $a,a+1,\dots ,a+t-1$ for some integer a. Their residues modulo t are pairwise distinct and therefore represent all elements of ${\mathbb{Z}}_t$. Hence, $\psi$ is also a modular H-irregular total k-labeling, and thus $\mathrm{mths}(G,H)\le k$.

On the other hand, (3) and the assumption $\mathrm{ths}(G,H)=k$ imply that $k=\mathrm{ths}(G,H)\le \mathrm{mths}(G,H)$. Therefore, $\mathrm{ths}(G,H)=\mathrm{mths}(G,H)=k$. □

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 6 we obtain the next corollary.

Corollary 4.

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 6 yields the following corollary.

Corollary 5.

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 6, ${\psi }_m$ 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 6.

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 equality holds in (3) for paths, fans and ladders. Moreover, in each case, the common value of $\mathrm{ths}(G,H)$ and $\mathrm{mths}(G,H)$ attains the lower bound given in Theorem 1, showing that this bound is sharp for these graph families.

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 7.

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.

In the calculations below, we repeatedly use the identity $⌈(A+D)/D⌉=⌈A/D⌉+1$, valid for every integer A and every positive integer D, together with the specified ranges of the indices.

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 6, we conclude that $\mathrm{mths}(R_n,R_m)=\mathrm{ths}(R_n,R_m)=k$. This completes the proof. □

The lower bound in Theorem 1 can be improved if all copies of H share some graph elements. The labels assigned to these common vertices and edges contribute the same amount to every H-weight and therefore cannot be used to distinguish the weights. This observation yields the following lower bound.

Theorem 8.

Let G be a graph containing exactly t subgraphs $H_1,H_2,\dots ,H_t$ isomorphic to H. Let S be a set of graph elements common to all the subgraphs $H_1,H_2,\dots ,H_t$, that is, $S\subseteq {\displaystyle \bigcap _{i=1}^t}\left(V\left(H_i\right)\cup E\left(H_i\right)\right)$. If $\left|V\right(H\left)\right|+\left|E\right(H\left)\right|-\left|S\right|>0$, then

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

Proof. 

Let $\psi$ be an H-irregular total k-labeling of G, and put $\sigma =\sum _{a\in S}\psi \left(a\right)$.

Each subgraph $H_i$ contains exactly $\left|V\right(H\left)\right|+\left|E\right(H\left)\right|-\left|S\right|$ graph elements outside S. Therefore, its weight is at least $\left|V\right(H\left)\right|+\left|E\right(H\left)\right|-\left|S\right|+\sigma$ and at most $\left(\left|V\right(H\left)\right|+\left|E\right(H\left)\right|-\left|S\right|\right)k+\sigma$. Since the t weights are pairwise distinct integers, the largest one is at least $\left|V\right(H\left)\right|+\left|E\right(H\left)\right|-\left|S\right|+\sigma +t-1$. Consequently,

$$\left|V\left(H\right)\right|+\left|E\left(H\right)\right|-\left|S\right|+\sigma +t-1\le \left(\left|V\right(H\left)\right|+\left|E\right(H\left)\right|-\left|S\right|\right)k+\sigma ,$$

which gives

$$k\ge ⌈1+\frac{t-1}{\left|V\right(H\left)\right|+\left|E\right(H\left)\right|-\left|S\right|}⌉.$$

□

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 9.

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. Put $k=⌈\left(6m+n-2\right)/\left(7m-2\right)⌉$. Since the vertex z belongs to all these subgraphs and $|V\left(L{W}_m\right)|+|E\left(L{W}_m\right)|=7m-1,$ Theorem 8, applied with $S=\left\{z\right\}$, gives

$$\mathrm{ths}(L{W}_n,L{W}_m)\ge ⌈1+\frac{n-m}{7m-2}⌉=⌈\frac{6m+n-2}{7m-2}⌉=k.$$

To prove the reverse inequality, it suffices to construct an $L{W}_m$-irregular total k-labeling of $L{W}_n$.

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

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

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 6, 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 10.

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 such that $2\le m\le n$. The strong ladder-wing graph $R{W}_n$ contains exactly $t=n-m+1$ subgraphs isomorphic to $R{W}_m$. Put $k=⌈\left(8m+n-4\right)/\left(9m-4\right)⌉$. Since the vertex z belongs to all these subgraphs and $|V\left(R{W}_m\right)|+|E\left(R{W}_m\right)|=9m-3$, Theorem 8, applied with $S=\left\{z\right\}$, gives

$$\mathrm{ths}(R{W}_n,R{W}_m)\ge ⌈1+\frac{n-m}{9m-4}⌉=⌈\frac{8m+n-4}{9m-4}⌉=k.$$

It remains to construct an $R{W}_m$-irregular total k-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:

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

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

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

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

$$\begin{array}{cc}\hfill w{t}_{{\psi }_m}\left(R{W}_m^1\right)=& {\displaystyle \sum _{i=1}^m}\left[{\psi }_m\left(x_i\right)+{\psi }_m\left(y_i\right)+{\psi }_m\left(x_i{y}_i\right)+{\psi }_m\left(x_{i}z\right)+{\psi }_m\left(y_{i}z\right)\right]+{\psi }_m\left(z\right)\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^{m-1}}\left[{\psi }_m\left(x_i{x}_{i+1}\right)+{\psi }_m\left(y_i{y}_{i+1}\right)+{\psi }_m\left(x_i{y}_{i+1}\right)+{\psi }_m\left(x_{i+1}{y}_i\right)\right]\hfill \\ \hfill =& {\displaystyle \sum _{i=1}^m}5+1+{\displaystyle \sum _{i=1}^{m-1}}4=5m+1+4(m-1)=9m-3.\hfill \end{array}$$

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 6 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. Irregularity Strengths with Respect to a Fixed Covering

The friendship graph provides an interesting example illustrating the difference between the classical and modular settings. Recall that a friendship graph ${\mathfrak{F}}_n$, $n\ge 1$, consists of n triangles sharing a common central vertex. Consequently, the number of subgraphs of ${\mathfrak{F}}_n$ isomorphic to ${\mathfrak{F}}_m$ 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. The following result gives infinite classes of friendship graphs for which the unrestricted modular labeling does not exist.

Corollary 7.

Let m and n be positive integers such that $1\le m\le n$. If $\left({\displaystyle \genfrac{}{}{0pt}{}{n}{m}}\right)\equiv 2\left(\mathrm{mod}4\right)$ and $\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{m-1}}\right)$ is even, then

$$\mathrm{mths}({\mathfrak{F}}_n,{\mathfrak{F}}_m)=\infty .$$

Proof. 

The graph ${\mathfrak{F}}_n$ contains $t=\left({\displaystyle \genfrac{}{}{0pt}{}{n}{m}}\right)$ subgraphs isomorphic to ${\mathfrak{F}}_m$. The common central vertex belongs to all t such subgraphs, whereas every other vertex and every edge of ${\mathfrak{F}}_n$ belongs to $\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{m-1}}\right)$ of them. Thus, all the corresponding numbers $c\left(x\right)$ are even, and the result follows from Theorem 5. □

In particular, if $m=2$ or $m=3$ and $n\equiv 5\left(\mathrm{mod}8\right)$, then

$$\mathrm{mths}({\mathfrak{F}}_n,{\mathfrak{F}}_m)=\infty ,$$

whereas $\mathrm{ths}({\mathfrak{F}}_n,{\mathfrak{F}}_m)$ is finite by Theorem 8 of [14].

The preceding nonexistence result shows that considering all copies of ${\mathfrak{F}}_m$ may prevent a modular labeling from existing. This motivates the covering-dependent variant, in which modular irregularity is required only with respect to the members of a fixed covering. We consider the fixed covering consisting of the copies induced by consecutive triangles.

Let $\mathcal{C}=\{H_1,H_2,\dots ,H_t\}$ be a fixed H-covering of a graph G. A total k-labeling $\psi$ is called an H-irregular total k-labeling with respect to $\mathcal{C}$ if the weights of the members of $\mathcal{C}$ are pairwise distinct. The minimum such k is called the total H-irregularity strength of G with respect to $\mathcal{C}$, and is denoted by ${\mathrm{ths}}_{\mathcal{C}}(G,H)$. If the function ${\lambda }_{\mathcal{C}}:\mathcal{C}\to {\mathbb{Z}}_t$, defined by

$${\lambda }_{\mathcal{C}}\left(H_i\right)\equiv w{t}_{\psi }\left(H_i\right)\left(\mathrm{mod}t\right)$$

is bijective, then $\psi$ is called a modular H-irregular total k-labeling with respect to $\mathcal{C}$. The minimum such k is called the modular total H-irregularity strength of G with respect to $\mathcal{C}$, and is denoted by ${\mathrm{mths}}_{\mathcal{C}}(G,H)$.

For $j=1,2,\dots ,n-m+1$, let ${\mathfrak{F}}_m^j$ denote the subgraph of ${\mathfrak{F}}_n$ 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 ${\mathfrak{F}}_m$-covering of ${\mathfrak{F}}_n$.

Theorem 11.

Let ${\mathfrak{F}}_n$, $n\ge 1$, be a friendship graph, and let m be a positive integer such that $1\le m\le n$. For the ${\mathfrak{F}}_m$-covering $\mathcal{C}=\{{\mathfrak{F}}_m^1,{\mathfrak{F}}_m^2,\dots ,{\mathfrak{F}}_m^{n-m+1}\}$ defined above,

$${\mathrm{mths}}_{\mathcal{C}}({\mathfrak{F}}_n,{\mathfrak{F}}_m)={\mathrm{ths}}_{\mathcal{C}}({\mathfrak{F}}_n,{\mathfrak{F}}_m)=⌈\frac{4m+n}{5m}⌉.$$

Proof. 

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

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

Let $\psi$ be an arbitrary ${\mathfrak{F}}_m$-irregular total k-labeling with respect to $\mathcal{C}$. Put $k=⌈\left(4m+n\right)/5m⌉$. Since the vertex z belongs to all subgraphs in $\mathcal{C}$ and $|V\left({\mathfrak{F}}_m\right)|+|E\left({\mathfrak{F}}_m\right)|=5m+1,$ the argument used in Theorem 8, applied to the members of $\mathcal{C}$ with $S=\left\{z\right\}$, gives

$${\mathrm{ths}}_{\mathcal{C}}({\mathfrak{F}}_n,{\mathfrak{F}}_m)\ge ⌈1+\frac{n-m}{5m}⌉=⌈\frac{4m+n}{5m}⌉=k.$$

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

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

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 ${\mathfrak{F}}_m$-weight of ${\mathfrak{F}}_m^j$ under ${\psi }_m$ is

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

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

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

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

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

Hence, the ${\mathfrak{F}}_m$-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 ${\mathfrak{F}}_m$-irregular total labeling and a modular ${\mathfrak{F}}_m$-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 that

$${\mathrm{mths}}_{\mathcal{C}}({\mathfrak{F}}_n,{\mathfrak{F}}_m)={\mathrm{ths}}_{\mathcal{C}}({\mathfrak{F}}_n,{\mathfrak{F}}_m)=⌈\frac{4m+n}{5m}⌉.$$

□

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.

Since the values of ${\mathrm{ths}}_{\mathcal{C}}(G,H)$ and ${\mathrm{mths}}_{\mathcal{C}}(G,H)$ may depend on the choice of $\mathcal{C}$, it would be interesting to study this dependence and to determine their minimum and maximum values over all H-coverings of G. Another problem is to characterize the coverings for which ${\mathrm{ths}}_{\mathcal{C}}(G,H)={\mathrm{mths}}_{\mathcal{C}}(G,H)$.

4. 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. We also obtained a necessary divisibility condition for the existence of a modular H-irregular total labeling. This condition shows that, in contrast to the classical parameter, the modular total H-irregularity strength may be infinite. As applications, we identified infinite classes of complete graphs and friendship graphs admitting no such modular labeling.

Using the sufficient condition, 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. We also introduced the corresponding covering-dependent invariants and determined their exact values for a fixed covering of friendship graphs. For each graph family for which we constructed a modular labeling, the obtained value attains the corresponding lower bound.

For the pairs $(n,m)$ not settled by Corollary 7, it remains open whether an unrestricted modular ${\mathfrak{F}}_m$-irregular total labeling of ${\mathfrak{F}}_n$ exists. If it exists, it would be interesting to determine whether $\mathrm{mths}({\mathfrak{F}}_n,{\mathfrak{F}}_m)=\mathrm{ths}({\mathfrak{F}}_n,{\mathfrak{F}}_m)$ or whether these two parameters can be different. More generally, an interesting problem is to characterize pairs $(G,H)$ for which $\mathrm{ths}(G,H)<\mathrm{mths}(G,H)<\infty .$ It would also be interesting to identify further graph classes for which the lower bound in Theorem 8 is sharp and to find additional structural conditions that either guarantee equality of the two parameters or prevent a modular H-irregular total labeling from existing.