Faraha Ashraf^a , Martin Bacaˇ b , Andrea Semanicov ˇ a-Fe ´ nov ˇ cˇ´ıkova´ b , Suhadi Wido Saputro^c

faraha27@gmail.com, martin.baca@tuke.sk, andrea.fenovcikova@tuke.sk, suhadi@math.itb.ac.id

A simple graph G = (V (G), E(G)) admits an H-covering if every edge in E(G) belongs to at least one subgraph of G isomorphic to a given graph H. A total k-labeling ϕ : V (G) ∪ E(G) → {1, 2, . . . , k} is called to be an H-irregular total k-labeling of the graph G admitting an H-covering if for every two different subgraphs H⁰ and H⁰⁰ isomorphic to H there is wtϕ(H⁰ ) 6= wtϕ(H⁰⁰), where wtϕ(H) = P v∈V (H) ϕ(v) + P e∈E(H) ϕ(e). The total H-irregularity strength of a graph G,

denoted by ths(G, H), is the smallest integer k such that G has an H-irregular total k-labeling. In this paper we determine the exact value of the cycle-irregularity strength of ladders and fan graphs.

Keywords: total H-irregular labeling, total cycle-irregularity strength, ladder, fan graph Mathematics Subject Classification : 05C78, 05C70

DOI: 10.5614/ejgta.2020.8.1.13

1. Introduction

Let G be a connected, simple and undirected graph with vertex set V (G) and edge set E(G). By a labeling we mean any mapping that maps a set of graph elements to a set of numbers (usually positive integers), called labels. If the domain is the vertex-set or the edge-set, the labelings are

Received: 31 August 2017, Revised: 26 January 2020, Accepted: 2 February 2020.

^aAbdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan

^bDepartment of Applied Mathematics and Informatics, Technical University, Letna 9, Ko ´ sice, Slovakia ˇ

^cFaculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jalan Ganesa No. 10 Bandung, Indonesia

called respectively vertex labelings or edge labelings. If the domain is V (G) ∪ E(G) then we call the labeling total labeling. The most complete recent survey of graph labelings is [12].

Baca, Jendrol ˇ ', Miller and Ryan in [9] defined the total labeling ϕ : V (G)∪E(G) → {1, 2, . . . , k} to be an edge irregular total k-labeling of the graph G if for every two different edges xy and x 0 y ⁰ of G one has

\[wt(xy) = \varphi(x) + \varphi(xy) + \varphi(y) \neq wt(x'y') = \varphi(x') + \varphi(x'y') + \varphi(y').\]

The total edge irregularity strength, tes(G), is defined as the minimum k for which G has an edge irregular total k-labeling.

Ivanco and Jendrol ˇ ' [14] posed a conjecture that for arbitrary graph G different from K⁵ and maximum degree ∆(G),

\[\operatorname{tes}(G) = \max\left\{ \left\lceil \frac{|E(G)|+2}{3} \right\rceil, \left\lceil \frac{\Delta(G)+1}{2} \right\rceil \right\}.\]

This conjecture has been verified for complete graphs and complete bipartite graphs in [15] and [16], for the Cartesian, categorical and strong products of two paths in [17, 3, 2], for the categorical product of two cycles in [4], for generalized Petersen graphs in [13], for generalized prisms in [10], for corona product of a path with certain graphs in [19] and for large dense graphs with (|E(G)| + 2)/3 ≤ (∆(G) + 1)/2 in [11].

There are several modifications of irregularity strength, namely the total vertex irregularity strength introduced in [9] and the edge irregularity strength introduced in [1]. In [20] there is confirmed the conjecture proposed by Nurdin, Baskoro, Salman and Gaos [18] for all trees with maximum degree five. The edge irregularity strength of some chain graphs is determined in [5].

An edge-covering of G is a family of subgraphs H1, H2, . . . , H^t such that each edge of E(G) belongs to at least one of the subgraphs Hⁱ , i = 1, 2, . . . , t. Then it is said that G admits an (H1, H2, . . . , Ht)-(edge) covering. If every subgraph Hⁱ is isomorphic to a given graph H, then the graph G admits an H-covering. Note, that in this case every subgraph isomorphic to H must be in the H-covering.

Let G be a graph admitting H-covering. For the subgraph H ⊆ G under the total k-labeling ϕ, we define the associated H-weight as

\[wt_{\varphi}(H) = \sum_{v \in V(H)} \varphi(v) + \sum_{e \in E(H)} \varphi(e).\]

A total k-labeling ϕ is called an H-irregular total k-labeling of the graph G if for every two different subgraphs H⁰ and H⁰⁰ isomorphic to H there is wtϕ(H⁰ ) 6= wtϕ(H⁰⁰). The total Hirregularity strength of a graph G, denoted ths(G, H), is the smallest integer k such that G has an H-irregular total k-labeling. If H is isomorphic to K2, then the K2-irregular total k-labeling is isomorphic to the edge irregular total k-labeling and thus the total K2-irregularity strength of a graph G is equivalent to the total edge irregularity strength, that is ths(G, K2) = tes(G).

Analogously we can define an H-irregular edge k-labeling and an H-irregular vertex k-labeling. For the subgraph H ⊆ G under the vertex k-labeling α, α : V (G) → {1, 2, . . . , k}, the

associated H-weight is defined as

\[wt_{\alpha}(H) = \sum_{v \in V(H)} \alpha(v)\]

and under the edge k-labeling \(\beta, \beta: E(G) \to \{1, 2, \dots, k\}\), we define the associated H-weight

\[wt_{\beta}(H) = \sum_{e \in E(H)} \beta(e).\]

A vertex k-labeling \(\alpha\) is called an H-irregular vertex k-labeling of the graph G if for every two different subgraphs H' and H'' isomorphic to H there is \(wt_{\alpha}(H') \neq wt_{\alpha}(H'')\). The vertex H-irregularity strength of a graph G, denoted by vhs(G,H), is the smallest integer k such that G has an H-irregular vertex k-labeling. Note, that \(vhs(G,H) = \infty\) if there exist two subgraphs in G isomorphic to H that have the same vertex sets. An edge k-labeling \(\beta\) is called an H-irregular edge k-labeling of the graph G if for every two different subgraphs H' and H'' isomorphic to H there is \(wt_{\beta}(H') \neq wt_{\beta}(H'')\). The edge H-irregularity strength of a graph G, denoted by ehs(G,H), is the smallest integer k such that G has an H-irregular edge k-labeling.

The notion of the vertex (edge) H-irregularity strength was introduced in [6]. The total H-irregularity strength was defined in [7] and its lower bound is given by the following theorem.

Theorem 1.1. [7] Let G be a graph admitting an H-covering given by t subgraphs isomorphic to H. Then

\[\operatorname{ths}(G, H) \ge \left[1 + \frac{t-1}{|V(H)| + |E(H)|}\right].\]

The precise value of the total H-irregularity strength of G-amalgamation of graphs is given in [8] and it proves that the lower bound in Theorem 1.1 is tight.

Let G be a graph admitting H-covering. By the symbol \(\mathbb{H}_m^S = (H_1^S, H_2^S, \dots, H_m^S)\), we denote the set of all subgraphs of G isomorphic to H such that the graph \(S, S \not\cong H\), is their maximum common subgraph. Thus \(V(S) \subset V(H_i^S)\) and \(E(S) \subset E(H_i^S)\) for every \(i=1,2,\dots,m\). The next theorem presented in [7] gives another lower bound of the total H-irregularity strength.

Theorem 1.2. [7] Let G be a graph admitting an H-covering. Let \(S_i\), i = 1, 2, ..., z, be all subgraphs of G such that \(S_i\) is a maximum common subgraph of \(m_i\), \(m_i \ge 2\), subgraphs of G isomorphic to H. Then

ths\[(G, H) \ge \max \left\{ \left\lceil 1 + \frac{m_1 - 1}{|V(H/S_1)| + |E(H/S_1)|} \right\rceil, \dots, \left\lceil 1 + \frac{m_z - 1}{|V(H/S_z)| + |E(H/S_z)|} \right\rceil \right\}.\]

In this paper we determine the exact value of the cycle-irregularity strength of ladders and fan graphs.

2. Total cycle-irregular labelings of ladders

Let \(L_n \cong P_n \square P_2\), \(n \geq 3\), be a ladder with the vertex set \(V(L_n) = \{v_i, u_i : i = 1, 2, ..., n\}\) and the edge set \(E(L_n) = \{v_i v_{i+1}, u_i u_{i+1} : i = 1, 2, ..., n - 1\} \cup \{v_i u_i : i = 1, 2, ..., n\}\).

In [7] is determined the exact value of the total cycle-irregularity strength of ladders when the cycle is either of length 4 or 6.

Theorem 2.1. [7] Let Lⁿ ∼= PnP2, n ≥ 3, be a ladder admitting a C2m-covering, m = 2, 3. Then

ths\[(L_n, C_{2m}) = \lceil \frac{3m+n}{4m} \rceil\].

In this section we extend the previous result for all feasible cycle-coverings.

Theorem 2.2. Let Lⁿ ∼= PnP2, n ≥ 3, be a ladder admitting a C2m-covering, 2 ≤ m ≤ d(n + 1)/2e. Then

ths\[(L_n, C_{2m}) = \lceil \frac{3m+n}{4m} \rceil\].

Proof. It is easy to see that the ladder Lⁿ ∼= PnP2, n ≥ 3, admits a C2m-covering for m = 2, 3, . . . , d(n + 1)/2e. Put k = 3m+n 4m . According to Theorem 1.1 k is the lower bound of ths(Ln, C2m). In order to show the converse inequality, it only remains to describe a C2m-irregular total k-labeling ϕ^m : V (Ln) ∪ E(Ln) → {1, 2, . . . , k} as follows

\[\varphi_m(v_i) = \left\lceil \frac{i+3m}{4m} \right\rceil \qquad \text{for } i = 1, 2, \dots, n,\] \[\varphi_m(u_i) = \begin{cases} \left\lceil \frac{i}{4m} \right\rceil & \text{for } i \equiv 0, 3m \pmod{4m}, i = 3m, 4m, 7m, 8m, \dots, n, \\ \left\lceil \frac{i+2m-1}{4m} \right\rceil & \text{for } i \not\equiv 0, 3m \pmod{4m}, i = 1, 2, \dots, n, \end{cases}\] \[\varphi_m(v_i v_{i+1}) = \left\lceil \frac{i+m}{4m} \right\rceil & \text{for } i = 1, 2, \dots, n-1,\] \[\varphi_m(u_i u_{i+1}) = \left\lceil \frac{i+2m}{4m} \right\rceil & \text{for } i = 1, 2, \dots, n-1,\] \[\varphi_m(v_i u_i) = \left\lceil \frac{i+2m}{4m} \right\rceil & \text{for } i = 1, 2, \dots, n.\]

We can see that all edge labels and vertex labels are at most k.

Every cycle C2^m in Lⁿ is of the form

\[C_{2m}^i = v_i v_{i+1} \dots v_{i+m-1} u_{i+m-1} u_{i+m-2} \dots u_i v_i,\]

where i = 1, 2, . . . , n − m + 1. It is easy to see that every edge of Lⁿ belongs to at least one cycle C i ^2m if m = 2, 3, . . . , d(n + 1)/2e.

For the C2m-weight of the cycle C i ²m, i = 1, 2, . . . , n − m + 1, under the total labeling ϕm, we get

\[wt_{\varphi_m}(C_{2m}^i) = \sum_{v \in V(C_{2m}^i)} \varphi_m(v) + \sum_{e \in E(C_{2m}^i)} \varphi_m(e)\] \[= \sum_{j=0}^{m-1} \varphi_m(v_{i+j}) + \sum_{j=0}^{m-1} \varphi_m(u_{i+j}) + \sum_{j=0}^{m-2} \varphi_m(v_{i+j}v_{i+j+1}) + \sum_{j=0}^{m-2} \varphi_m(u_{i+j}u_{i+j+1})\] \[+ \varphi_m(v_iu_i) + \varphi_m(v_{i+m-1}u_{i+m-1})\] \[(1)\]

and for the C2m-weight of the cycle C i+1 ^2m , i = 1, 2, . . . , n − m, we obtain

\[wt_{\varphi_m}(C_{2m}^{i+1}) = \sum_{v \in V(C_{2m}^{i+1})} \varphi_m(v) + \sum_{e \in E(C_{2m}^{i+1})} \varphi_m(e)\]

\[= \sum_{j=1}^{m} \varphi_m(v_{i+j}) + \sum_{j=1}^{m} \varphi_m(u_{i+j}) + \sum_{j=1}^{m-1} \varphi_m(v_{i+j}v_{i+j+1}) + \sum_{j=1}^{m-1} \varphi_m(u_{i+j}u_{i+j+1}) + \varphi_m(v_{i+1}u_{i+1}) + \varphi_m(v_{i+m}u_{i+m}).\] (2)

Now we count the difference between the C2m-weights of the cycle C i+1 ^2m and C i ^2m for i = 1, 2, . . . , n − m. According to (1) and (2) we get

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Let us distinguish four cases.

Case 1. i ≡ 0 (mod 4m)

For the difference of weights of cycles we get

\[wt_{\varphi_m}(C_{2m}^{i+1}) - wt_{\varphi_m}(C_{2m}^i) = \left\lceil \frac{i+1+2m}{4m} \right\rceil + \left\lceil \frac{i+2m-1}{4m} \right\rceil + \left\lceil \frac{i+4m}{4m} \right\rceil + \left\lceil \frac{i+3m}{4m} \right\rceil + \left\lceil \frac{i+3m-1}{4m} \right\rceil + \left\lceil \frac{i+m}{4m} \right\rceil - \left\lceil \frac{i+3m}{4m} \right\rceil - \left\lceil \frac{i+3m}{4m} \right\rceil - \left\lceil \frac{i+1}{4m} \right\rceil - \left\lceil \frac{i+3m-1}{4m} \right\rceil - \left\lceil \frac{i+3m-1}{4m} \right\rceil = \left\lceil \frac{i+2m+1}{4m} \right\rceil + \left\lceil \frac{i+2m-1}{4m} \right\rceil + 1 - \left\lceil \frac{i+2m}{4m} \right\rceil - \left\lceil \frac{i+1}{4m} \right\rceil.\]

Since i = 4mt, t = 1, 2, . . . , thus

\[\begin{split} wt_{\varphi_m}(C_{2m}^{i+1}) - wt_{\varphi_m}(C_{2m}^i) &= \left\lceil \frac{4mt + 2m + 1}{4m} \right\rceil + \left\lceil \frac{4mt + 2m - 1}{4m} \right\rceil + 1 - \left\lceil \frac{4mt + 2m}{4m} \right\rceil - \left\lceil \frac{4mt + 1}{4m} \right\rceil \\ &= t + \left\lceil \frac{2m + 1}{4m} \right\rceil + t + \left\lceil \frac{2m - 1}{4m} \right\rceil + 1 - t - \left\lceil \frac{2m}{4m} \right\rceil - t - \left\lceil \frac{1}{4m} \right\rceil = 1. \end{split}\]

Case 2. i ≡ 2m (mod 4m)

For the difference of weights of cycles we get

\[wt_{\varphi_m}(C_{2m}^{i+1}) - wt_{\varphi_m}(C_{2m}^i) = \left\lceil \frac{i+1+2m}{4m} \right\rceil + \left\lceil \frac{i+2m-1}{4m} \right\rceil + \left\lceil \frac{i+4m}{4m} \right\rceil + \left\lceil \frac{i+3m}{4m} \right\rceil + \left\lceil \frac{i+m}{4m} \right\rceil + \left\lceil \frac{i+m}{4m} \right\rceil - \left\lceil \frac{i+3m}{4m} \right\rceil - \left\lceil \frac{i+3m}{4m} \right\rceil - \left\lceil \frac{i+2m-1}{4m} \right\rceil - \left\lceil \frac{i+1}{4m} \right\rceil - \left\lceil \frac{i+3m-1}{4m} \right\rceil = \left\lceil \frac{i+2m+1}{4m} \right\rceil + 1 + \left\lceil \frac{i}{4m} \right\rceil + \left\lceil \frac{i+m}{4m} \right\rceil - \left\lceil \frac{i+2m}{4m} \right\rceil - \left\lceil \frac{i+1}{4m} \right\rceil - \left\lceil \frac{i+3m-1}{4m} \right\rceil.\]

For i = 4mt + 2m, t = 1, 2, . . . , we get

\[\begin{split} wt_{\varphi_m}(C_{2m}^{i+1}) - wt_{\varphi_m}(C_{2m}^i) &= \left\lceil \frac{4mt + 2m + 2m + 1}{4m} \right\rceil + 1 + \left\lceil \frac{4mt + 2m}{4m} \right\rceil + \left\lceil \frac{4mt + 2m + m}{4m} \right\rceil \\ &- \left\lceil \frac{4mt + 2m + 2m}{4m} \right\rceil - \left\lceil \frac{4mt + 2m + 1}{4m} \right\rceil - \left\lceil \frac{4mt + 2m + 3m - 1}{4m} \right\rceil \\ &= t + 1 + \left\lceil \frac{1}{4m} \right\rceil + 1 + t + \left\lceil \frac{2m}{4m} \right\rceil + t + \left\lceil \frac{3m}{4m} \right\rceil - t - 1 - t - \left\lceil \frac{2m + 1}{4m} \right\rceil \\ &- t - 1 - \left\lceil \frac{m - 1}{4m} \right\rceil = 1. \end{split}\]

Case 3. i ≡ 3m (mod 4m)

Now

\[wt_{\varphi_m}(C_{2m}^{i+1}) - wt_{\varphi_m}(C_{2m}^i) = \left\lceil \frac{i+1+2m}{4m} \right\rceil + \left\lceil \frac{i+2m-1}{4m} \right\rceil + \left\lceil \frac{i+4m}{4m} \right\rceil + \left\lceil \frac{i+3m}{4m} \right\rceil + \left\lceil \frac{i+m}{4m} \right\rceil + \left\lceil \frac{i+m}{4m} \right\rceil\]

\[\begin{split} &-\left\lceil\frac{i+3m}{4m}\right\rceil-\left\lceil\frac{i+m}{4m}\right\rceil-\left\lceil\frac{i+2m}{4m}\right\rceil-\left\lceil\frac{i}{4m}\right\rceil-\left\lceil\frac{i+1}{4m}\right\rceil-\left\lceil\frac{i+3m-1}{4m}\right\rceil\\ &=\left\lceil\frac{i+2m+1}{4m}\right\rceil+\left\lceil\frac{i+2m-1}{4m}\right\rceil+1+\left\lceil\frac{i+m}{4m}\right\rceil-\left\lceil\frac{i+2m}{4m}\right\rceil-\left\lceil\frac{i+1}{4m}\right\rceil\\ &-\left\lceil\frac{i+3m-1}{4m}\right\rceil\,. \end{split}\]

Since i = 4mt + 3m, t = 1, 2, . . . , it follows

\[wt_{\varphi_m}(C_{2m}^{i+1}) - wt_{\varphi_m}(C_{2m}^i) = \left\lceil \frac{4mt + 3m + 2m + 1}{4m} \right\rceil + \left\lceil \frac{4mt + 3m + 2m - 1}{4m} \right\rceil + 1 + \left\lceil \frac{4mt + 3m + m}{4m} \right\rceil - \left\lceil \frac{4mt + 3m + 2m}{4m} \right\rceil - \left\lceil \frac{4mt + 3m + 1}{4m} \right\rceil - \left\lceil \frac{4mt + 3m + 3m - 1}{4m} \right\rceil = t + 1 + \left\lceil \frac{m + 1}{4m} \right\rceil + t + 1 + \left\lceil \frac{m - 1}{4m} \right\rceil + 1 + t + 1 - t - 1 - \left\lceil \frac{m}{4m} \right\rceil - t - \left\lceil \frac{3m + 1}{4m} \right\rceil - t - 1 - \left\lceil \frac{2m - 1}{4m} \right\rceil = 1.\]

Case 4. i 6≡ 0, 2m, 3m (mod 4m)

In this case for the difference of weights of cycles we obtain

\[wt_{\varphi_m}(C_{2m}^{i+1}) - wt_{\varphi_m}(C_{2m}^i) = \left\lceil \frac{i+1+2m}{4m} \right\rceil + \left\lceil \frac{i+2m-1}{4m} \right\rceil + \left\lceil \frac{i+4m}{4m} \right\rceil + \left\lceil \frac{i+3m}{4m} \right\rceil + \left\lceil \frac{i+3m-1}{4m} \right\rceil + \left\lceil \frac{i+m}{4m} \right\rceil - \left\lceil \frac{i+2m}{4m} \right\rceil - \left\lceil \frac{i+2m}{4m} \right\rceil - \left\lceil \frac{i+2m-1}{4m} \right\rceil - \left\lceil \frac{i+1}{4m} \right\rceil - \left\lceil \frac{i+3m-1}{4m} \right\rceil = \left\lceil \frac{i+2m+1}{4m} \right\rceil + \left\lceil \frac{i+4m}{4m} \right\rceil - \left\lceil \frac{i+2m}{4m} \right\rceil - \left\lceil \frac{i+1}{4m} \right\rceil.\]

Let i = 4mt + s, t = 0, 1, 2, . . . and 1 ≤ s ≤ 4m − 1, s 6= 2m, 3m. Then we have

\[wt_{\varphi_m}(C_{2m}^{i+1}) - wt_{\varphi_m}(C_{2m}^i) = \left\lceil \frac{4mt + s + 2m + 1}{4m} \right\rceil + \left\lceil \frac{4mt + s + 4m}{4m} \right\rceil - \left\lceil \frac{4mt + s + 2m}{4m} \right\rceil - \left\lceil \frac{4mt + s + 1}{4m} \right\rceil\]\[= t + \left\lceil \frac{s + 2m + 1}{4m} \right\rceil + t + 1 + \left\lceil \frac{s}{4m} \right\rceil - t - \left\lceil \frac{s + 2m}{4m} \right\rceil - t - \left\lceil \frac{s + 1}{4m} \right\rceil\]\[= \left\lceil \frac{s + 2m + 1}{4m} \right\rceil + \left\lceil \frac{s}{4m} \right\rceil - \left\lceil \frac{s + 2m}{4m} \right\rceil - \left\lceil \frac{s + 1}{4m} \right\rceil + 1.\]

If 1 ≤ s ≤ 2m − 1 then

\[\left\lceil \frac{s+2m+1}{4m} \right\rceil = 1, \quad \left\lceil \frac{s}{4m} \right\rceil = 1, \quad \left\lceil \frac{s+2m}{4m} \right\rceil = 1 \quad \text{and} \quad \left\lceil \frac{s+1}{4m} \right\rceil = 1.\]

If 2m + 1 ≤ s ≤ 3m − 1 or 3m + 1 ≤ s ≤ 4m − 1 then

\[\left\lceil \frac{s+2m+1}{4m} \right\rceil = 2, \quad \left\lceil \frac{s}{4m} \right\rceil = 1, \quad \left\lceil \frac{s+2m}{4m} \right\rceil = 2 \quad \text{and} \quad \left\lceil \frac{s+1}{4m} \right\rceil = 1.\]

We can see that for every value of parameter s

\[wt_{\varphi_m}(C_{2m}^{i+1}) - wt_{\varphi_m}(C_{2m}^i) = 1.\]

Previous cases prove that the labeling ϕ^m is the desired C2m-irregular total k-labeling of Ln. This concludes the proof.

3. Total cycle-irregular labelings of fan graphs

A fan graph Fn, n ≥ 2, is a graph obtained by joining all vertices of a path Pⁿ to a further vertex. Thus Fⁿ contains n+1 vertices, say, u, v1, v2, . . . , vⁿ and 2n−1 edges uvⁱ , i = 1, 2, . . . , n, and vivi+1, i = 1, 2, . . . , n − 1.

In [7] was given the exact value of the total C3-irregularity strength of the fan graph Fn.

Theorem 3.1. [7] Let Fn, n ≥ 2, be a fan graph on n + 1 vertices. Then

ths\[(F_n, C_3) = \left\lceil \frac{n+3}{5} \right\rceil\].

The next theorem completes this result for arbitrary cycle-covering.

Theorem 3.2. Let Fⁿ be a fan graph on n + 1 vertices, n ≥ 2 and 3 ≤ m ≤ d(n + 3)/2e. Then

ths\[(F_n, C_m) = \left\lceil \frac{n+m}{2m-1} \right\rceil\].

Proof. Clearly, for every m, 3 ≤ m ≤ d(n + 3)/2e, the fan graph Fⁿ admits a Cm-covering with exactly n−m+ 2 cycles Cm. In view of the lower bound from Theorem 1.2 it suffices to prove the existence of a Cm-irregular total labeling ϕ : V (Fn) ∪ E(Fn) → {1, 2, . . . , d(n + m)/(2m − 1)e} such that wtϕ(C j ^m) 6= wtϕ(C i ^m) for every i, j = 1, 2, . . . , n − m + 2, j 6= i. We describe the irregular total labeling ϕ^m in the following way

\[\varphi_m(u) = 1,\] \[\varphi_m(v_i) = \begin{cases} \left\lceil \frac{i+2}{2m-1} \right\rceil & \text{for } i \not\equiv m+1 \pmod{(2m-1)}, i=1,2,\ldots,n, \\ \left\lceil \frac{i+2}{2m-1} \right\rceil + 1 & \text{for } i \equiv m+1 \pmod{(2m-1)}, i=m+1,3m,\ldots,n, \end{cases}\] \[\varphi_m(v_i v_{i+1}) = \begin{cases} \left\lceil \frac{i+m}{2m-1} \right\rceil & \text{for } i \not\equiv m+1,2m-3,2m-1 \pmod{(2m-1)}, \\ i=1,2,\ldots,n, \end{cases}\] \[\varphi_m(v_i u) = \begin{cases} \left\lceil \frac{i+m}{2m-1} \right\rceil - 1 & \text{for } i \equiv m+1,2m-3,2m-1 \pmod{(2m-1)}, \\ i=m+1,2m-3,2m-1,3m,4m-4,4m-2,\ldots,n, \end{cases}\] \[\varphi_m(v_i u) = \begin{cases} \left\lceil \frac{i+m}{2m-1} \right\rceil & \text{for } i \not\equiv m+1,2m-2 \pmod{(2m-1)}, i=1,2,\ldots,n, \\ \left\lceil \frac{i+m}{2m-1} \right\rceil - 1 & \text{for } i \equiv m+1,2m-2 \pmod{(2m-1)}, \\ i=m+1,2m-2,3m,4m-3,\ldots,n. \end{cases}\]

Evidently, every edge label and vertex label is not greater than d(n + m)/(2m − 1)e. Every cycle C^m in Fⁿ is of the form

\[C_m^i = v_i v_{i+1} \dots v_{i+m-2} u v_i,\]

where i = 1, 2, . . . , n − m + 2.

For the Cm-weight of the cycle C i ^m, i = 1, 2, . . . , n − m + 2, under the total labeling ϕm, we get

\[wt_{\varphi_m}(C_m^i) = \sum_{v \in V(C_m^i)} \varphi_m(v) + \sum_{e \in E(C_m^i)} \varphi_m(e)\] \[= \sum_{j=0}^{m-2} \varphi_m(v_{i+j}) + \varphi_m(u) + \sum_{j=0}^{m-3} \varphi_m(v_{i+j}v_{i+j+1}) + \varphi_m(v_iu) + \varphi_m(v_{i+m-2}u)\](3)

and for the Cm-weight of the cycle C i+1 ^m , i = 1, 2, . . . , n − m + 1, we obtain

\[wt_{\varphi_m}(C_m^{i+1}) = \sum_{v \in V(C_m^{i+1})} \varphi_m(v) + \sum_{e \in E(C_m^{i+1})} \varphi_m(e)\] \[= \sum_{j=1}^{m-1} \varphi_m(v_{i+j}) + \varphi_m(u) + \sum_{j=1}^{m-2} \varphi_m(v_{i+j}v_{i+j+1}) + \varphi_m(v_{i+1}u) + \varphi_m(v_{i+m-1}u).\](4)

Now we count the difference between the Cm-weights of the cycle C i+1 ^m and C i ^m for i = 1, 2, . . . , n − m + 1. According to (3) and (4) we get

\[\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\]

Let us distinguish nine cases.

Case 1. \[i \equiv 2 \pmod{(2m-1)}\], i.e., \(i = 2 + (2m-1)t\), \(t = 0, 1, ...\), then

\[\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\]

Case 2. i ≡ 3 (mod (2m − 1)), i.e., i = 3 + (2m − 1)t, t = 0, 1, . . . , then we get

\[wt_{\varphi_m}(C_m^{i+1}) - wt_{\varphi_m}(C_m^i) = \varphi_m(v_{m+2+(2m-1)t}) + \varphi_m(v_{m+1+(2m-1)t}v_{m+2+(2m-1)t})\] \[+ \varphi_m(v_{m+2+(2m-1)t}u) + \varphi_m(v_{4+(2m-1)t}u) - \varphi_m(v_{3+(2m-1)t})\] \[- \varphi_m(v_{3+(2m-1)t}v_{4+(2m-1)t}) - \varphi_m(v_{3+(2m-1)t}u)\] \[- \varphi_m(v_{m+1+(2m-1)t}u)\] \[= \left\lceil \frac{m+4+(2m-1)t}{2m-1} \right\rceil + \left\lceil \frac{2m+1+(2m-1)t}{2m-1} \right\rceil - 1 + \left\lceil \frac{2m+2+(2m-1)t}{2m-1} \right\rceil\] \[+ \left\lceil \frac{m+4+(2m-1)t}{2m-1} \right\rceil - \left\lceil \frac{5+(2m-1)t}{2m-1} \right\rceil - \left\lceil \frac{m+3+(2m-1)t}{2m-1} \right\rceil\] \[- \left\lceil \frac{m+3+(2m-1)t}{2m-1} \right\rceil - \left\lceil \frac{2m+1+(2m-1)t}{2m-1} \right\rceil + 1\] \[= (1+t) + (2+t) - 1 + (2+t) + (1+t) - (1+t)\]

\[-(1+t) - (1+t) - (2+t) + 1 = 1.\]

\[\begin{aligned} \text{\it Case 3. } i &\equiv m-1 \pmod{(2m-1)}, \text{ i.e., } i = m-1 + (2m-1)t, \, t = 0, 1, \dots, \text{ then we get} \\ wt_{\varphi_m}(C_m^{i+1}) - wt_{\varphi_m}(C_m^i) &= \varphi_m(v_{2m-2+(2m-1)t}) + \varphi_m(v_{2m-3+(2m-1)t}v_{2m-2+(2m-1)t}) \\ &+ \varphi_m(v_{2m-2+(2m-1)t}u) + \varphi_m(v_{m+(2m-1)t}u) \\ &- \varphi_m(v_{m-1+(2m-1)t}) - \varphi_m(v_{m-1+(2m-1)t}v_{m+(2m-1)t}) \\ &- \varphi_m(v_{m-1+(2m-1)t}u) - \varphi_m(v_{2m-3+(2m-1)t}u) \\ &= \left\lceil \frac{2m+(2m-1)t}{2m-1} \right\rceil + \left\lceil \frac{3m-3+(2m-1)t}{2m-1} \right\rceil - 1 + \left\lceil \frac{3m-2+(2m-1)t}{2m-1} \right\rceil \\ &- 1 + \left\lceil \frac{2m+(2m-1)t}{2m-1} \right\rceil - \left\lceil \frac{m+1+(2m-1)t}{2m-1} \right\rceil - \left\lceil \frac{2m-1+(2m-1)t}{2m-1} \right\rceil \\ &- \left\lceil \frac{2m-1+(2m-1)t}{2m-1} \right\rceil - \left\lceil \frac{3m-3+(2m-1)t}{2m-1} \right\rceil \\ &= (2+t) + (2+t) - 1 + (2+t) - 1 + (2+t) - (1+t) \\ &- (1+t) - (1+t) - (2+t) = 1. \end{aligned}\]

Case 4. i ≡ m (mod (2m − 1)), i.e., i = m + (2m − 1)t, t = 0, 1, . . . In this case holds

\[\begin{split} wt_{\varphi_m}(C_m^{i+1}) - wt_{\varphi_m}(C_m^i) &= \varphi_m(v_{2m-1+(2m-1)t}) + \varphi_m(v_{2m-2+(2m-1)t}v_{2m-1+(2m-1)t}) \\ &+ \varphi_m(v_{2m-1+(2m-1)t}u) + \varphi_m(v_{m+1+(2m-1)t}u) - \varphi_m(v_{m+(2m-1)t}) \\ &- \varphi_m(v_{m+(2m-1)t}v_{m+1+(2m-1)t}) - \varphi_m(v_{m+(2m-1)t}u) \\ &- \varphi_m(v_{2m-2+(2m-1)t}u) \\ &= \left\lceil \frac{2m+1+(2m-1)t}{2m-1} \right\rceil + \left\lceil \frac{3m-2+(2m-1)t}{2m-1} \right\rceil + \left\lceil \frac{3m-1+(2m-1)t}{2m-1} \right\rceil \\ &+ \left\lceil \frac{2m+1+(2m-1)t}{2m-1} \right\rceil - 1 - \left\lceil \frac{m+2+(2m-1)t}{2m-1} \right\rceil - \left\lceil \frac{2m+(2m-1)t}{2m-1} \right\rceil \\ &- \left\lceil \frac{2m+(2m-1)t}{2m-1} \right\rceil - \left\lceil \frac{3m-2+(2m-1)t}{2m-1} \right\rceil + 1 \\ &= (2+t) + (2+t) + (2+t) + (2+t) - 1 - (1+t) - (2+t) \\ &- (2+t) - (2+t) + 1 = 1. \end{split}\]

Case 5. i ≡ m + 1 (mod (2m − 1)), i.e., i = m + 1 + (2m − 1)t, t = 0, 1, . . . , thus

\[wt_{\varphi_m}(C_m^{i+1}) - wt_{\varphi_m}(C_m^i) = \varphi_m(v_{2m+(2m-1)t}) + \varphi_m(v_{2m-1+(2m-1)t}v_{2m+(2m-1)t}) + \varphi_m(v_{2m+(2m-1)t}u) + \varphi_m(v_{m+2+(2m-1)t}u) - \varphi_m(v_{m+1+(2m-1)t}) - \varphi_m(v_{m+1+(2m-1)t}v_{m+2+(2m-1)t}) - \varphi_m(v_{m+1+(2m-1)t}u) - \varphi_m(v_{2m-1+(2m-1)t}u) = \left[\frac{2m+2+(2m-1)t}{2m-1}\right] + \left[\frac{3m-1+(2m-1)t}{2m-1}\right] - 1 + \left[\frac{3m+(2m-1)t}{2m-1}\right] + 1 - \left[\frac{2m+2+(2m-1)t}{2m-1}\right] + 1 - \left[\frac{2m+1+(2m-1)t}{2m-1}\right] + 1 - \left[\frac{3m-1+(2m-1)t}{2m-1}\right]\]

\[=(2+t)+(2+t)-1+(2+t)+(2+t)-(1+t)-1\]\[-(2+t)+1-(2+t)+1-(2+t)=1.\]

Case 6. \[i \equiv 2m - 3 \pmod{(2m-1)}\], i.e., \(i = 2m - 3 + (2m-1)t\), \(t = 0, 1, ...\), thus

\[wt_{\varphi_m}(C_m^{i+1}) - wt_{\varphi_m}(C_m^i) = \varphi_m(v_{3m-4+(2m-1)t}) + \varphi_m(v_{3m-5+(2m-1)t}v_{3m-4+(2m-1)t}) + \varphi_m(v_{3m-4+(2m-1)t}u) + \varphi_m(v_{2m-2+(2m-1)t}u) - \varphi_m(v_{2m-3+(2m-1)t}) - \varphi_m(v_{2m-3+(2m-1)t}v_{2m-2+(2m-1)t}) - \varphi_m(v_{2m-3+(2m-1)t}u) - \varphi_m(v_{3m-5+(2m-1)t}u) = \left[\frac{3m-2+(2m-1)t}{2m-1}\right] + \left[\frac{4m-5+(2m-1)t}{2m-1}\right] + \left[\frac{4m-4+(2m-1)t}{2m-1}\right] + \left[\frac{3m-2+(2m-1)t}{2m-1}\right] - \left[\frac{3m-3+(2m-1)t}{2m-1}\right] - \left[\frac{3m-3+(2m-1)t}{2m-1}\right] + 1 - \left[\frac{3m-3+(2m-1)t}{2m-1}\right] - \left[\frac{4m-5+(2m-1)t}{2m-1}\right] = (2+t) + (2+t) + (2+t) + (2+t) - 1 - (1+t) - (2+t) + 1 - (2+t) - (2+t) = 1.\]

Case 7. \[i \equiv 2m-2 \pmod{(2m-1)}\], i.e., \(i = 2m-2+(2m-1)t\), \(t = 0, 1, \ldots\), thus

\[\begin{split} wt_{\varphi_m}(C_m^{i+1}) - wt_{\varphi_m}(C_m^i) &= \varphi_m(v_{3m-3+(2m-1)t}) + \varphi_m(v_{3m-4+(2m-1)t}v_{3m-3+(2m-1)t}) \\ &+ \varphi_m(v_{3m-3+(2m-1)t}u) + \varphi_m(v_{2m-1+(2m-1)t}u) \\ &- \varphi_m(v_{2m-2+(2m-1)t}) - \varphi_m(v_{2m-2+(2m-1)t}v_{2m-1+(2m-1)t}) \\ &- \varphi_m(v_{2m-2+(2m-1)t}u) - \varphi_m(v_{3m-4+(2m-1)t}u) \\ &= \left\lceil \frac{3m-1+(2m-1)t}{2m-1} \right\rceil + \left\lceil \frac{4m-4+(2m-1)t}{2m-1} \right\rceil + \left\lceil \frac{4m-3+(2m-1)t}{2m-1} \right\rceil \\ &+ \left\lceil \frac{3m-1+(2m-1)t}{2m-1} \right\rceil - \left\lceil \frac{2m+(2m-1)t}{2m-1} \right\rceil - \left\lceil \frac{3m-2+(2m-1)t}{2m-1} \right\rceil \\ &- \left\lceil \frac{3m-2+(2m-1)t}{2m-1} \right\rceil + 1 - \left\lceil \frac{4m-4+(2m-1)t}{2m-1} \right\rceil \\ &= (2+t)+(2+t)+(2+t)+(2+t)-(2+t) \\ &- (2+t)-(2+t)+1-(2+t)=1. \end{split}\]

Case 8. \[i \equiv 2m - 1 \pmod{(2m - 1)}\], i.e., \(i = 2m - 1 + (2m - 1)t\), \(t = 0, 1, ...\), thus

\[\begin{split} wt_{\varphi_m}(C_m^{i+1}) - wt_{\varphi_m}(C_m^i) = & \varphi_m(v_{3m-2+(2m-1)t}) + \varphi_m(v_{3m-3+(2m-1)t}v_{3m-2+(2m-1)t}) \\ & + \varphi_m(v_{3m-2+(2m-1)t}u) + \varphi_m(v_{2m+(2m-1)t}u) \\ & - \varphi_m(v_{2m-1+(2m-1)t}) - \varphi_m(v_{2m-1+(2m-1)t}v_{2m+(2m-1)t}) \\ & - \varphi_m(v_{2m-1+(2m-1)t}u) - \varphi_m(v_{3m-3+(2m-1)t}u) \\ = & \left\lceil \frac{3m+(2m-1)t}{2m-1} \right\rceil + \left\lceil \frac{4m-3+(2m-1)t}{2m-1} \right\rceil + \left\lceil \frac{4m-2+(2m-1)t}{2m-1} \right\rceil \\ & + \left\lceil \frac{3m+(2m-1)t}{2m-1} \right\rceil - \left\lceil \frac{2m+1+(2m-1)t}{2m-1} \right\rceil - \left\lceil \frac{3m-1+(2m-1)t}{2m-1} \right\rceil + 1 \end{split}\]

\[-\left\lceil \frac{3m-1+(2m-1)t}{2m-1} \right\rceil - \left\lceil \frac{4m-3+(2m-1)t}{2m-1} \right\rceil\]
=(2+t)+(2+t)+(2+t)+(2+t)-(2+t)
-(2+t)+1-(2+t)-(2+t)=1.

Case 9. i 6≡ 2, 3, m − 1, m, m + 1, 2m − 3, 2m − 2, 2m − 1 (mod (2m − 1)). Then

\[wt_{\varphi_m}(C_m^{i+1}) - wt_{\varphi_m}(C_m^i) = \varphi_m(v_{i+m-1}) + \varphi_m(v_{i+m-2}v_{i+m-1}) + \varphi_m(v_{i+m-1}u) + \varphi_m(v_{i+1}u)\] \[- \varphi_m(v_i) - \varphi_m(v_iv_{i+1}) - \varphi_m(v_iu) - \varphi_m(v_{i+m-2}u)\] \[= \left\lceil \frac{(i+m-1)+2}{2m-1} \right\rceil + \left\lceil \frac{(i+m-2)+m}{2m-1} \right\rceil + \left\lceil \frac{(i+m-1)+m}{2m-1} \right\rceil + \left\lceil \frac{(i+1)+m}{2m-1} \right\rceil\] \[- \left\lceil \frac{i+2}{2m-1} \right\rceil - \left\lceil \frac{i+m}{2m-1} \right\rceil - \left\lceil \frac{i+m}{2m-1} \right\rceil - \left\lceil \frac{i+2}{2m-1} \right\rceil + 1.\] \[= 2 \left\lceil \frac{i+m+1}{2m-1} \right\rceil - 2 \left\lceil \frac{i+m}{2m-1} \right\rceil + \left\lceil \frac{i}{2m-1} \right\rceil - \left\lceil \frac{i+2}{2m-1} \right\rceil + 1.\]

Now we distinguish three subcases.

If \[i = 1 + (2m - 1)t\], \(t = 0, 1, ...\), then

\[wt_{\varphi_m}(C_m^{i+1}) - wt_{\varphi_m}(C_m^i) = 2\left\lceil \frac{1 + (2m-1)t + m + 1}{2m-1} \right\rceil - 2\left\lceil \frac{1 + (2m-1)t + m}{2m-1} \right\rceil + \left\lceil \frac{1 + (2m-1)t}{2m-1} \right\rceil - \left\lceil \frac{1 + (2m-1)t + 2}{2m-1} \right\rceil + 1 = 2(1+t) - 2(1+t) + (1+t) - (1+t) + 1\] \[= 1.\]

If \[i = s + (2m - 1)t\], \(t = 0, 1, ...\) and \(4 \le s \le m - 2\), then

\[wt_{\varphi_m}(C_m^{i+1}) - wt_{\varphi_m}(C_m^i) = 2\left\lceil \frac{s + (2m-1)t + m + 1}{2m-1} \right\rceil - 2\left\lceil \frac{s + (2m-1)t + m}{2m-1} \right\rceil + \left\lceil \frac{s + (2m-1)t}{2m-1} \right\rceil - \left\lceil \frac{s + (2m-1)t + 2}{2m-1} \right\rceil + 1 = 2(1+t) - 2(1+t) + (1+t) - (1+t) + 1\] \[= 1.\]

If i = s + (2m − 1)t, t = 0, 1, . . . and m + 2 ≤ s ≤ 2m − 4, in this case we get

\[wt_{\varphi_m}(C_m^{i+1}) - wt_{\varphi_m}(C_m^i) = 2\left\lceil \frac{s + (2m-1)t + m + 1}{2m-1} \right\rceil - 2\left\lceil \frac{s + (2m-1)t + m}{2m-1} \right\rceil + \left\lceil \frac{s + (2m-1)t}{2m-1} \right\rceil - \left\lceil \frac{s + (2m-1)t + 2}{2m-1} \right\rceil + 1 = 2(2+t) - 2(2+t) + (1+t) - (1+t) + 1\] \[= 1.\]

Thus, according to all these cases we get that

\[wt_{\varphi_m}(C_m^{i+1}) - wt_{\varphi_m}(C_m^i) = 1\]

for every i, i = 1, 2, . . . , n − m + 1. This concludes the proof.

4. Conclusion

In this paper we determined the exact value of the cycle-irregularity strength of ladders and fan graphs. We proved that for the ladder Lⁿ ∼= PnP2, n ≥ 3, admitting a C2m-covering, 2 ≤ m ≤ d(n+ 1)/2e, ths(Ln, C2m) = 3m+n 4m . Moreover, for the fan graph Fⁿ on n+ 1 vertices, n ≥ 2 and 3 ≤ m ≤ d(n + 3)/2e, ths(Fn, Cm) = n+m 2m−1 .

For the edge (vertex) cycle-irregularity strength of ladders was proved the following.

Theorem 4.1. [6] Let Lⁿ ∼= PnP2, n ≥ 2, be a ladder. Then

\[\operatorname{ehs}(L_n, C_4) = \left\lceil \frac{n+2}{4} \right\rceil.\]

Theorem 4.2. [6] Let Lⁿ ∼= PnP2, n ≥ 3, be a ladder. Let m be a positive integer, m ≤ d(n + 1)/2e. Then

\[\operatorname{vhs}(L_n, C_{2m}) = \left\lceil \frac{m+n}{2m} \right\rceil.\]

In [6] is also given the exact value for the vertex cycle-irregularity strength for fan graphs.

Theorem 4.3. [6] Let Fⁿ be a fan graph on n+ 1 vertices, n ≥ 2 and 3 ≤ m ≤ d(n + 3)/2e. Then

\[\operatorname{vhs}(F_n, C_m) = \left\lceil \frac{n}{m-1} \right\rceil.\]

According to results proved in [6] it is needed to find the edge cycle-irregularity strength for ladders and fans for every feasible length of cycles. We suppose that these parameters equal to the lower bounds. We conclude the paper with the following conjectures.

Conjecture 1. Let Lⁿ ∼= PnP2, n ≥ 2, be a ladder admitting a C2m-covering, 3 ≤ m ≤ d(n + 1)/2e. Then

\[\operatorname{ehs}(L_n, C_{2m}) = \left\lceil \frac{m+n}{2m} \right\rceil.\]

Conjecture 2. Let Fⁿ be a fan graph on n + 1 vertices, n ≥ 2 and 3 ≤ m ≤ d(n + 3)/2e. Then

\[\operatorname{ehs}(F_n, C_m) = \left\lceil \frac{n+1}{m} \right\rceil.\]

Acknowledgement

The research for this article was supported by APVV-15-0116, by VEGA 1/0233/18 and by Riset P3MI 1016/I1.C01/PL/2017.

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