Graph labeling places symbols on the structural elements of a graph in order to encode combinatorial information. Depending on which elements receive labels, one distinguishes vertex labelings, edge labelings, and total labelings where both vertices and edges are assigned values [1]. The modern development of labeling theory goes back to the foundational work of Rosa [2] and subsequent contributions by Golomb on graph numbering [3].

In this article, we focus on antimagic edge labelings. Let be a finite graph with a set of vertices V(G) and an edge set E(G). Fix a bijection f : . For each vertex , set the sum of the labels on the edges incident with v. We call f an antimagic edge labeling if for all distinct vertices . A graph G is antimagic if it admits such a labeling. For brevity, tG denotes the disjoint union of t copies of G.

The formal notion of antimagic graphs itself was introduced by Hartsfield and Ringel [4]. Beyond their intrinsic combinatorial interest, antimagic labelings have found use in cryptography. They have been incorporated into block-cipher style constructions and secure communication schemes, illustrating how discrete labeling constraints can be leveraged for difusion and uniqueness properties. For instance, Prihandini and Adawiyah designed a Hill-cipher variant informed by antimagic principles [5]; Gurjar and Krishnaa surveyed several antimagic labeled families with a view toward cryptographic frameworks [6] and Gondalia discussed implications for secure data transfer [7].

A long-standing conjecture due to Hartsfield and Ringel asserts that every tree other than is antimagic [4]. Although a full resolution is not yet available, substantial progress has been made for broad subfamilies. Kaplan, Lev, and Roditty showed that any nontrivial rooted tree in which every non-leaf vertex has at least two children is antimagic [8]. Liang, Wong, and Zhu showed that if a tree T has no vertices of degree two, then its subdivision , obtained by inserting a new vertex on every edge, is antimagic [9]. For caterpillars, Lozano et al. established that a caterpillar of order n is [10]. For any caterpillar whose spine has order , or with p prime, Wong and Zhu proved that it is 1-antimagic [11]. For caterpillars whose maximum degree is three, Deng and Li showed the graphs are antimagic [12].

Disconnected structures have also been investigated as natural extensions of the connected case. Shang et al. analyzed star forests and identified antimagic conditions when there is no component and at most one component is [13]. It was proved by Chang et al. that each double spider admits an antimagic labeling [14] and shown by Sierra et al. that any forest with no more than one vertex of degree two is antimagic [15]. Additional families and techniques appear in [16, 17, 18, 19, 20]; for a broad survey of labeling results, including antimagic labelings, see Gallian [21].

Together, these developments provide a robust platform for studying more elaborate constructions. In particular, they motivate the present work on disjoint unions that combine familiar tree classes with 4-cycles, and on constructive labeling frameworks that yield antimagic behavior across wide parameter ranges. The results below aim to widen the catalogue of antimagic graphs and to supply flexible methods that apply simultaneously to trees and small even cycles.

In this paper, we first investigate the antimagic labeling of unions of 3-paths, . To establish the notation used in labeling , it is relatively straightforward and follows naturally from the set of integers assigned to its edges. We provide an independent proof of the following result, which has also been previously established in [22].

Proposition 2.1.The graph is not antimagic for

Proof. Let denote the degree-2 vertex in the i-th copy of Assume on the contrary that has an antimagic labeling f. Without loss of generality, assume that the edges incident to v are labeled with 1 and . This means that the vertex sum of is . If , then which means that in the labeling , an edge incident to receives the label . This implies that the vertex sum of the pendant vertex of this edge is a contradiction. Hence, assume that . This means that the remaining edges of are labeled with

Without loss of generality, assume that the edges incident to are labeled with 2 and . That is, the vertex sum of is . Now , otherwise the vertex sum of v is , which is also the vertex sum of in ϕ, a contradiction. Since , then . But this implies that, in an edge incident to is labeled with , which means that the vertex sum of the pendant vertex adjacent to is , a contradiction. This completes the proof. □

Let be a star which is also a complete bipartite graph for any positive integer n.

Proposition 2.2. where and , then G is antimagic.

Proof. We label the edges of the k-th with the integers for as shown in the Figure 1. The vertex sums of are given by . The vertex sums of G are all unique, which implies that G is antimagic. □

Theorem 2.3 ([13, Theorem 2.1]). If G is a union of stars containing no S1 and at most one S2, then G is antimagic.

Proposition 2.2 is a specific case within the broader scope of Theorem 2.3 proved by [13]. However, the two results use diferent labeling methods. Figure 2 shows antimagic labeling for , Figure 2(a) applies the approach described in Proposition 2.2, while Figure 2(b) follows the method outlined in the proof of Theorem 2.3.

Definition 2.4.A linear forest is defined as a collection of disjoint paths, each containing more than one vertex.

A linear forest refers to a linear forest in which none of its components is a path of order In 2018, Shang [23] established the following theorem concerning linear forests, providing significant insights into their properties and structure.

Theorem 2.5 ([23, Theorem 2.8]). ree linear forests are antimagic.

Let P be a linear forest and where is the k-th path of order for all k. Let be the vertices in sequential order of for where and are the pendant vertices at each end of the path. Let be the edges between the vertices and for . Let , where According to Theorem 2.5, f serves as an antimagic labeling of . Moreover, the first and last edges of the path are assigned labels from the smallest integers 2t in the set , as determined by the equations provided below:

Definition 2.6.A broom is a graph of n vertices, which has a path P with d vertices and pendant vertices, all of these being adjacent to either the origin u or the terminus v of the path P [24]. A broom forest is a disjoint union of broom graphs. denotes the k-th broom, then the single broom forest is defined as , where t is the number of brooms in the forest.

Proposition 2.7.A single broom forest is antimagic for

Proof. Let be a function such that , and , for Then, the vertex sums are given by , and . Since the vertex sums of B are distinct, B is antimagic. □

Theorem 2.8.A single broom forest is antimagic for and

Proof. Suppose that a single broom forest B is constructed from a linear forest , where is antimagic. Let be an antimagic labeling that satisfies (1). Now let be a function such that and for . Note that for and it is clear that

Note that the vertex sum in (2) is all distinct and the largest vertex sum is 6t. Since is antimagic, in (3) are all diferent. Hence, B is antimagic. □

Figure 3 illustrates an example of an antimagic labeling of single broom forests.

Definition 2.9.A double broom graph is obtained from a path with vertices by attaching s pendant (leaf) vertices to the left end vertex and attaching s pendant vertices to the right end vertex equivalently, its vertex set is

and its edge set is

Hence, has vertices and edges [25]. A double broom forest is a disjoint union of double brooms. denotes the k-th double broom, then the forest is defined as , where t is the number of double brooms in the forest.

An example of an antimagic labeling for double broom forests is shown in the Figure 4.

Proposition 2.10.A double broom forest is antimagic for

Proof. Let be a function such that Then the vertex sums of D given by are all distinct. Therefore, D is antimagic. □

Proposition 2.11.A double broom forest is antimagic for

Proof. Let be a function defined as 2 for Based on this labeling, the corresponding vertex sums in D are: . All of these vertex sums are pairwise distinct. Hence, D is antimagic. □

Theorem 2.12.A double broom forest is antimagic and

Proof. Suppose that the double broom forest D is constructed from a linear forest where is antimagic. Let be an antimagic labeling that satisfies (1). Now, let } be a function defined as , additionally, for . Note that for . The vertex sums of D are given by

We can see that the vertex sums in (4) are all distinct and the largest vertex sum is . Since is antimagic, in (5) are all diferent. Hence, D is antimagic. □

The double broom is a tree characterized by exactly two vertices of degree 3. This construction can be generalized by joining two disjoint copies of the star through a single edge. The resulting tree, denoted , consists of 2n vertices and edges. In this structure, the two central vertices, each initially serving as the center of an , have degree , while all remaining vertices have degree 1.

Proposition 2.13. is antimagic for the integers and

Proof. Since consists of two copies of , we first label the edge that joins the two copies and then label the edges within each . For the k-th copy of , we assign the edge connecting the two copies of the label , while the edges within the two copies of are labeled with the integers and for . As a result, all vertex sums in are unique, given by the set . Therefore, 1) is antimagic. □

Theorem 2.14. is antimagic for the integers and

Proof. For the k-th copy of , we first label with integers and with integers , respectively. The edge connecting and is assigned with the label , for . It can be shown that all the vertex sums of are distinct, so we conclude that is antimagic. □

We investigate antimagic labelings of disjoint unions of 4-cycles . A labeling of a cycle is written in the form , where and d are distinct and represent the edge labels encountered when traversing the cycle cyclically, either clockwise or counterclockwise. Throughout this section, we use the notation to denote a labeled and to denote the labeling of a path For example, the labeling is illustrated in Figure 5.

Proposition 3.1.If where , then is antimagic.

Proof. We label the edges of k-th with the integers for . The vertex sums of are given by . Since all the vertex sums of G are distinct, G is antimagic. □

Proposition 2.1 establishes that the graph is not antimagic for However, when considering the disjoint union , We observe that G does admit an antimagic labeling. In what follows, we present all possible antimagic labelings for the graph , as summarized in Table 1. Furthermore, Table 2 illustrates the diferent configurations in which distinct positive integers can be assigned to the edges of the graph , while preserving the antimagic property.

Proposition 3.2.Let for the positive integer . Then G is antimagic.

Proof. We label the only graph as (1, 2) and note that the vertex sums for the path are {1, 2, 3}. For the k-th , the edges are labeled with 1) for The vertex sums of the 4-cycles are given by . Since all the vertex sums of G are distinct, G is antimagic. □

Proposition 3.3.Let G be a graph consisting of unions of paths or cycles where n is a positive integer with . If G is antimagic, then is also antimagic.

Proof. Suppose that where is a positive integer. Given that G is antimagic, there is a one-to-one correspondence between the and the label set such that holds for any two distinct vertices u and v. We label the edges of with integers . In the graph the largest edge label (if it exists) is , which is less than , the smallest edge label in . Hence, is antimagic. □

Referring to Theorem 2.5 and Proposition 3.3, we can immediately derive the following results.

Proposition 3.4.Let be a , P -free linear forest. Then is antimagic for a positive integer

Definition 3.5.An extended Skolem sequence of order n is a sequence of length 2n + 1, consisting of integers, that satisfies the lowing conditions:

(i) For each integer , the sequence contains exactly two occurrences

(ii) If the two positions satisfy then

(iii) There exists exactly one position i such that

An alternative representation of an extended Skolem sequence, as introduced in [26], is as a set of ordered pairs , such that the union of all indices equals the entire index set Abrham and Kotzig [27] proposed a connection between extended Skolem sequences and a specific type of additive permutation, called -permutations. In their work, they demonstrated that each extended Skolem sequence corresponds uniquely to a σ-permutation.

The subsequent theorems will establish the application of extended Skolem sequences of order n in constructing antimagic labelings for the graph .

Theorem 3.6.Given with , the graph G is antimagic.

Proof. For any integer , we examine the extended Skolem sequence, where each element is a triple of the form , with

We assign edge labels to the k-th copy of using the pair , for each . It is evident that the maximum label among all the components is

To construct the labels for the components, we define the set , which in this case simplifies to

This set has cardinality . Since each contains 4 edges and we must assign a distinct label to each edge, the number of elements in the set used for labeling must be a multiple of 4. To satisfy this divisibility condition, we augment Y depending on the residue class of q modulo 4 as follows:

(i) If (mod 4), let ;

(ii) If q ≡ 1 (mod 4), let

(iii) I

(iv) If q ≡ 3 (mod 4), let

We then arrange the elements of the relevant set in increasing order and denote them by . The edge labels for are assigned as , for . Since all edge labels are distinct and each vertex sum is unique, the resulting graph G is antimagic. □

Theorem 3.7.The graph is antimagic for every integer

Proof. We use the extended Skolem sequence for the integer , defined by the triple for

To label the edges of each component, we associate to the k-th path the pair for all . The largest value among these labels is clearly , which will serve as an upper bound for the labeling of the paths.

For the labeling of the components, we introduce the set , which yields for the values considered. This set contains distinct integers. To ensure that the total number of labels used for the cycles is a multiple of four, we extend Y by appending two additional integers greater than , namely and , and define the augmented set , which has size . We now arrange the elements of in increasing order, denoted as For each , the edge labels of j-th are assigned in the specific pattern , ensuring a varied distribution of values across each 4-cycle. Given that all edge labels used across and components are distinct, and each vertex of the graph receives a unique induced sum, we conclude that the resulting graph G is antimagic. □

Theorem 3.8.For all integers , the graph is antimagic.

Proof. To construct the extended Skolem sequence for any integer , we use the triple

label the edges of each component, we associate to the k-th path the label pair for every . Among all such labels, the maximum value assigned is

We define the set to construct the labels for the components. This yields , a set containing exactly 3q elements. Since each 4-cycle requires four distinct edge labels, the set used for labeling must be divisible by 4 in size. To achieve this when (mod 4), we extend the set by appending additional integers greater than according to the value of modulo 4. Specifically, when (mod , the set already contains a multiple of 4 elements and needs no adjustment. If (mod 4), we define (mod 4), we take ; and for , we let In each case, the augmented set has cardinality divisible by 4. Once the appropriate set is determined based on the residue of we arrange its elements in increasing order as . The edge labels for the are then assigned using the pattern for each . As all edge labels across the and components are distinct, and each vertex sum is unique, it follows that the resulting graph G is antimagic. □

Theorem 3.9.Let be a positive integer. Then the graph is antimagic.

Proof. Consider the extended Skolem sequence for integers , represented by the triple for each

We assign edge labels to each component as follows. For all the k-th copy of is labeled with the pair . It follows that the maximum value used among the labels of the components is . We now define a set to be used in labeling the components. . It is straightforward to verify that . Furthermore, the union of all labels assigned to the components and the set , covers the entire set

To ensure that the number of labels used for the components is divisible by we extend the set depending on the parity of If is even, we define , so that is divisible by 4. If is odd, we define , again ensuring that is divisible by 4.

We begin by arranging the elements of if is even, or if is odd, in increasing order, denoted as . The edge labels for the sequence of 4-cycles are then assigned according to the following rule: the first 4-cycle is labeled with the quadruple , while the second is labeled with . For each subsequent integer k ranging from 2 to , the th 4-cycle is labeled with the quadruple . Since the induced vertex sums are all distinct, the graph G is antimagic. □

From Theorems 3.6, 3.7, 3.8 and 3.9, we have the following result.

Theorem 3.10.The graph is antimagic for every integer

Proposition 3.11.Let with and suppose that . Then the graph is antimagic labeling.

Proof. The result is clear from Theorem 3.10 and Proposition 3.3.

Proposition 3.12. is antimagic for positive integers and

Proof. We label the graph according to the method described in the proof of Theorem 2.8, using the integers . The edges of the s disjoint 4- cycles are then labeled with the integers . Specifically, for each , the edges of the k-th are assigned the labels . Note that , where is the unique vertex of degree 3 in . Since 6 is strictly less than all vertex sums in the components, all vertex sums in the graph are distinct. Therefore, G is antimagic. □

Proposition 3.13.Let for positive integers and . Then G is antimagic.

Proof. In Section 2, we established that is antimagic for . The graph contains exactly two vertices of degree 3, each incident to three edges. We assign the integers 1, 2, 3, 4, 5, 6 to these six edges, ensuring that each receives a distinct label. Consequently, the vertex sums at the two degree-3 vertices are strictly less than . For each , we label the edges of the k-th 4-cycle with the values . All of these labels are all greater than those used in , and they ensure distinct vertex sums throughout the entire graph. Therefore, the union is antimagic. □

Proposition 3.14. is antimagic for integers and

Proof. We begin by labeling the star with the integers . The vertex of degree in has a vertex sum . Next, we assign labels to the edges of each in using the set for The smallest possible vertex sum in the components is . For , we have and for and the value does not coincide with any vertex sum arising from the components. Therefore, all vertex sums in the graph are distinct, implying that G is antimagic. □

Proposition 3.15.Let and be integers. is antimagic { \frac { 5 + { \sqrt { 1 + 6 4 t } } } { 2 } } \end{document} ]]>

Proof. We begin by labeling the edges of the star with integers For each , we assign edge labels to the i-th 4-cycle using the values . This labeling ensures that the largest vertex sum in each is . Hence, the maximum vertex sum across all t copies of , denoted , is . Meanwhile, the maximum vertex sum in the star is . To ensure that the entire graph is antimagic, meaning that all vertex sums are distinct, it must satisfy the inequality 2 n + 8 t - 3 \end{document} ]]> . Solving this yields the condition { \frac { 5 + { \sqrt { 1 + 6 4 t } } } { 2 } } \end{document} ]]> □

In this paper, we investigated antimagic edge labelings for disjoint unions of sparse graphs, focusing on unions of paths, star– and broom–type trees, and on the role of adjoining 4-cycles. Our results combine sharp obstructions with scalable constructive labelings.

We first established a definitive limitation for linear forests: the disjoint union is not antimagic for every . Beyond this exceptional family, we gave explicit labeling schemes showing that broad classes of forests are antimagic, including unions of stars and several families of broom forests (single and double brooms, as well as joined-star variants), by partitioning the label set into ranges and controlling induced vertex sums.

We then extended the framework to graphs containing . We showed that the unions of 4-cycles admit antimagic labelings and that adding blocks can preserve and produce antimagicness in disjoint unions. In particular, the dificult case of many -components becomes tractable once suficiently many are adjoined: using extended Skolem sequences we derived suficient conditions for graphs of the form , including an explicit bound on s in terms of Finally, we demonstrated that the same cycle-block strategy interacts efectively with broom and star components, yielding further antimagic families with clear parameter ranges.