Group-antimagic Labelings of Multi-cyclic Graphs

Let A be a non-trivial abelian group. A connected simple graph G = (V,E) is A-antimagic if there exists an edge labeling f : E(G) → A\{0} such that the induced vertex labeling f+ : V (G) → A, defined by f+(v) = Σ {f(u, v) : (u, v) ∈ E(G)}, is a one-to-one map. The integer-antimagic spectrum of a graph G is the set IAM(G) = {k : G is Zk-antimagic and k ≥ 2}. In this paper, we analyze the integer-antimagic spectra for various classes of multi-cyclic graphs.


Introduction
A labeling of a graph is defined to be an assignment of values to the vertices and/or edges of the graph.Graph labeling is a very diverse and active field of study.A dynamic survey [6] maintained by Gallian contains over 1400 references to research papers and books on the topic.
Let G be a connected simple graph.For any non-trivial abelian group A (written additively), let A * = A\{0}, where 0 is the additive identity of A (sometimes denoted by 0 A ). Let a function f : E(G) → A * be an edge labeling of G. Any such labeling induces a map If there exists such an edge labeling f whose induced map f + on V (G) is one-to-one, we say that f is an A-antimagic labeling and that G is an A-antimagic graph.The integer-antimagic spectrum of a graph G is the set IAM(G) = {k : G is Z k -antimagic and k ≥ 2}.
A trivial lower bound for the least element of IAM(G) is the order of G; however, this is not always achieved, as seen in the following result from [1].Lemma 1.1 (Chan et al.).A graph of order 4m + 2, for all m ∈ N, is not Z 4m+2 -antimagic.
Motivation for our current work is found in the following conjecture.
Conjecture 1.1.Let G be a connected simple graph.If t is the least positive integer such that G is Z t -antimagic, then IAM(G) = {k : k ≥ t}.
A result of Jones and Zhang [7] finds the minimum element of IAM(G) for all connected graphs on 3 or more vertices.In their paper, a Z n -antimagic labeling of a graph on n vertices is referred to as a nowhere-zero modular edge-graceful labeling.This is a variation of a graceful labeling (originally called a β-valuation) which was introduced by Rosa [23] in 1967.The result is as follows, where the terminology has been adapted to better suit this paper.In [1], Conjecture 1.1 was shown to be true for various classes of graphs.The purpose of this paper is to provide additional evidence for Conjecture 1.1 by verifying it for various classes of multi-cyclic graphs.We use constructive methods to determine integer-antimagic spectra of the graph classes in question.

Some Known Results
In this section, we include some known results [1] for reference.In particular, theorems (with an included proof) are used in the construction of new Z k -antimagic labelings in this paper.Theorem 2.1.P 4m+r and C 4m+r , for all m ∈ N, are Z k -antimagic, for all k ≥ 4m + r if r = 0, 1, 3. P 4m+2 and C 4m+2 , for all m ∈ N, are Z k -antimagic, for all k ≥ 4m + 3.
Proof.Let e 1 , e 2 , . . ., e n−1 be edges of P n , from left to right.A Z k -antimagic labeling of P n can be obtained as follows.

Z k -antimagic Labelings of Wheels and Wheel-like Graphs
Let W n denote the wheel on n spokes, which is the graph containing a cycle of length n with another special vertex not on the cycle, called the central vertex, that is adjacent to every vertex on the cycle.Name the vertices of W n as follows: the central vertex is named v 0 and the other vertices are named counter-clockwise as v 1 , . . ., v n .We will refer to edges of the form v 0 v i for 1 ≤ i ≤ n as spokes and edges of the form v i v i+1 for 1 ≤ i ≤ n − 1 or v n v 1 as outer-cycle edges.The subgraph of W n formed by the outer-cycle edges will be referred to as the outer-cycle.Following the naming for the edges of a cycle found in the proof of Theorem 2.1, an outer-cycle edge v i v i+1 receives the name e i+1 , and the edge v n v 1 is named e 1 .Furthermore, for every i ̸ = 0 the spoke with end-vertex v i receives the name e ′ i .First, we note that W 2 ∼ = C 3 is clearly not Z 3 -antimagic.Figure 1 illustrates Z k -antimagic labelings (k ≥ 4), for W 2 and W 3 .
Proof.Case 1. Labeling W 4m+r for r = 0, 2: Let k ≥ 4m + r + 1 be fixed.The outer-cycle is of even length, and hence admits a one-factorization into two one-factors, say M 1 and M 2 .We will first define our labeling, f , on the outer-cycle edges as follows: The subgraph formed by the spokes is a star with 4m + r + 1 vertices, and hence has a Z k -antimagic labeling, g, by Theorem 2.2.Define the labeling on the spokes as f (e) = g(e) for all spokes e.This induces a labeling, Case 2. Labeling W 4m+3 : Let k ≥ 4m + 4 be fixed.We will first define a labeling, g, on the outer-cycle edges to be the labeling defined for a cycle of length 4m + 3 in Theorem 2.1.Notice that the labels induced by g on the vertices in the outer-cycle, denoted by g + , form the set {3, 4, . . ., 4m + 4, 4m + 5}.Now, define f : E(W 4m+3 ) → Z * k such that for every outer-cycle edge, e i , we have f (e i ) = g(e i ), and for the spokes: Thus, The labels on the vertices induced by f form the set {4, 5, . . ., 4m + 6, 4m + 7}; thus, f is the desired Z k -antimagic labeling.Case 3. Labeling W 4m+1 : Let k ≥ 4m + 3 be fixed.We will first define a labeling, g, on the outer-cycle edges to be the labeling defined for a cycle of length 4m + 1 in Theorem 2.1.Notice that the labels induced by g on the vertices in the outer-cycle, denoted by g + , form the set {2, 3, . . ., 4m + 1, 4m + 2}.Now, define f : E(W 4m+1 ) → Z * k such that for every outer-cycle edge, e i , we have f (e i ) = g(e i ), and for the spokes: Thus, The labels on the vertices induced by f form the set {3, 4, . . ., 4m+2, 4m+3, 4m+5}; thus, f is the desired Z k -antimagic labeling.Let F n denote the fan on n + 1 vertices, which is defined to be the graph obtained from W n by deleting the edge e 1 .We have the same vertex and edge names for F n as we did for W n (omitting the edge name e 1 , of course).Concerning F n , the subgraph formed by the edges of the form e i will be referred to as the outer-path, and its edges will be referred to as outer-path edges.Edges of the form e ′ i will still be referred to as spokes.
Proof.Case 1. Labeling F 4m+1 : Let k ≥ 4m+3 be fixed.Let g be the labeling on the outer-path edges obtained from the labeling of a cycle of length 4m + 1 in Theorem 2.1 with the edge e 1 omitted.Notice that the labels induced by g on the vertices in the outer-path, denoted by g + , form the set {1, 2, 4, 5, . . ., 4m + 2}.Now, define f : E(F 4m+1 ) → Z * k such that for every outer-path edge, e i , we have f (e i ) = g(e i ), and for the spokes: Thus, The labels on the vertices induced by f form the set {2, 4, 5, . . ., 4m + 4}; thus, f is the desired Z k -antimagic labeling.
Case 2. Labeling F 4m+3 : Let k ≥ 4m+4 be fixed.Let g be the labeling on the outer-path edges obtained from the labeling of a cycle of length 4m + 3 in Theorem 2.1 with the edge e 1 omitted.
Notice that the labels induced by g on the vertices in the outer-path, denoted by g + , form the set {2, 3, 5, 6, . . ., 4m + 5}.Now, define f : E(F 4m+3 ) → Z * k such that for every outer-path edge, e i , we have f (e i ) = g(e i ), and for the spokes: Thus, The labels on the vertices induced by f form the set {4, 5, . . ., 4m + 7}; thus, f is the desired Z k -antimagic labeling.Case 3. Labeling F 4m : Let k ≥ 4m + 1 be fixed.Let g be the labeling on the outer-path edges obtained from the labeling of a cycle of length 4m in Theorem 2.1 with the edge e 1 omitted.Notice that the labels induced by g on the vertices in the outer-path, denoted by g + , form the set {2, 3, 5, 6, . . ., 4m + 2}.Now, define f : E(F 4m ) → Z * k such that for every outer-path edge, e i , we have f (e i ) = g(e i ), and for the spokes: Thus, The labels on the vertices induced by f form the set {4, 5, . . ., 4m + 4}; thus, f is the desired Z k -antimagic labeling.Case 4. Labeling F 4m+2 : Let k ≥ 4m + 3 be fixed.First, re-label the edges of the outer-path with the assignment e i → e i−1 .Let g be the labeling on the outer-path edges obtained from the labeling of a path on 4m + 2 vertices in Theorem 2.1.Notice that the labels induced by g on the vertices in the outer-path, denoted by g + , form the set {1, 3, 4, . . ., 4m + 3}.Now, define f : E(F 4m+2 ) → Z * k such that for every outer-path edge, e i , we have f (e i ) = g(e i ), and for the spokes: Thus, The labels on the vertices induced by f form the set {3, 4, . . ., 4m + 5}; thus, f is the desired Z k -antimagic labeling.A friendship graph is a simple graph in which any two distinct vertices have exactly one common neighbor.A result of Erdős et al. [5] shows that all friendship graphs are isomorphic to some W n with a 1-factor deleted from the outer-cycle (thus, n must be even).Let F G n denote the friendship graph on n + 1 vertices.We name the vertices of F G n in the same way that we named the vertices of W n .We will refer to edges of the form v 0 v i for 1 ≤ i ≤ n, named e ′ i , as spokes and edges of the form v i v i+1 for i = 1, 3, 5, . . ., n − 1, named e (i+1)/2 , as outer 1-factor edges.The subgraph of F G n formed by the outer 1-factor edges will be referred to as the outer 1-factor.
For the remainder of the proof, we assume that n ≥ 4 and n is even.
Let k ≥ n + 2 be fixed.The subgraph formed by the spokes is a star with n edges, and therefore admits a Z k -antimagic labeling g (with central vertex having induced label 0 (mod k)), by Theorem 2.2.There must be some element x ∈ Z * k that g doesn't assign to any spoke.Now, define f : E(F G n ) → Z * k such that for every spoke, e ′ i , we have f (e ′ i ) = g(e ′ i ), and for every outer 1-factor edge, e i , we have f (e i ) = k − x.Let f + denote the labels induced by f on the vertices of Since all of the g(e ′ i )'s are distinct, so are the labels induced by f on the vertices of the outer 1-factor.Furthermore, we have that for all 1 ≤ i ≤ n, f + (v i ) ̸ ≡ 0 (mod k), otherwise there would be some i for which g(e ′ i ) = x.Since f + (v 0 ) ≡ 0 (mod k), f is the desired Z k -antimagic labeling.Now, let k = n + 1. Define the labeling f : E(F G n ) → Z * k such that for every spoke, e ′ i , we have f (e ′ i ) = i, and for the outer 1-factor edges: The labels induced on the vertices of the outer 1-factor by f form the set {3, 4, 5, . . ., n} ∪ {n+2, n+3}, and f + (v 0 ) ≡ 0 (mod n+1).Thus, f is the desired Z k -antimagic labeling.A helm on 2n + 1 vertices, denoted H n , is the graph obtained from the wheel graph, W n , by adjoining a pendant vertex to each vertex on the outer-cycle.In H n , the names of all vertices and edges in the subgraph isomorphic to W n have the same names as in W n , except that the edges e ′ i are referred to as inner-spokes.For each vertex v i , we name the leaf that is adjacent to it w i and refer to these vertices as pendant vertices.Each edge of the form v i w i is named e ′′ i , and these edges are referred to as outer-spokes.If we delete the outer-cycle edges of n , then we are left with a tree rooted at v 0 with n vertex-disjoint paths of length two attached to it.Denote this graph by H ′ n .It will be helpful to first define a Z k -antimagic labeling of H ′ n .We adopt the same names for vertices and edges in this graph as we have already defined for the underlying helm.
Proof.Let n be an even positive integer, and let k ≥ 2n + 1 be fixed.Define the function f : E(H ′ n ) → Z * k on the inner-spokes as follows.
Define f on the outer-spokes as follows.
Case 1. n even: Let k ≥ 2n + 1 be fixed.Define the function f : For the edges of the subgraph H ′ n , we define f the same as the Z k -antimagic labeling given in the proof of Lemma 3.4.The edges in the set E(H n ) \ E(H ′ n ) form a cycle of length n.Since n is even, we can label the outer-cycle edges by alternating 1 and k − 1.Thus, the labels on the outer-cycle edges contribute k to each induced vertex label f + (v i ).It follows that f is the desired Z k -antimagic labeling.

Case 2. n odd:
Let k ≥ 2n+1 be fixed.Notice that the outer-cycle can be viewed as a path on n+1 vertices in which the first and last vertices are identified.In order to label the edges of the outer-cycle, we first consider a path on n + 1 vertices.Label the edges of the f Now, we check that the induced vertex labels are all distinct.Notice that {f + (v i ) : 2 } = {2, 3, . . ., n + 3, n + 4}.We also have that {f Putting all of these sets of induced vertex labels together we have {f + (x) : all of the induced vertex labels are distinct.Thus, f is the desired Z k -antimagic labeling.

Z k -antimagic Labelings of the Square of Paths
The kth power of a graph G, denoted G k , is a graph with the same vertex set as G and two vertices are adjacent in G k if and only if their distance in G is at most k.

Figure 7 :
Figure 7: A Z k -antimagic labeling of H 2 for k ≥ 5, and a Z k -antimagic labeling of H 3 for k ≥ 7.

3. Theorem 2.2. Let
n ≥ 4 and S n denote the star graph having n − 1 leaves.If n is odd, then S n is Z k -antimagic, for all k ≥ n.Otherwise, S n is Z k -antimagic, for all k ≥ n + 2; but not Z n -antimagic nor Z n+1 -antimagic.
4.1.Let n ≥ 4. If n ≡ 2 (mod 4), then P 2 n is Z k -antimagic for all k ≥ n + 1.Otherwise, P 2 n is Z k -antimagic for all k ≥ n.Proof.Let G = P 2 n .Suppose the vertices of P n (from left to right) are v 1 , v 2 , . . ., v n and the edges of P n (from left to right) are e 1 , e 2 , . . ., e n−1 .
Label C G (n) and P n in the same way, as found in Case 1. Now, overlay the labelings of P n and C G (n), by identifying the vertices and edges (and adding their values).All of the vertex labels of G are distinct (mod k).We need to check if edges e 1 and e n−1 are still non-zero (mod k).Edge e i was initially labeled with one of the following: i+1 2 , if i is odd and 1≤ i ≤ 2m − 1; otherwise i+3 2 , if i is odd and 2m + 1 ≤ i ≤ 4m + 1. Adding 1 to either i+1 2 or i+3 2 yield non-zero values (mod k), for all k ≥ n = 4m + 3. Thus, P 2 n is Z k -antimagic for all k ≥ n.Case 3. n = 4m: Let C G (n) be the n-cycle v 1 v 3 v 5 • • • v n−1 v n v n−2 v n−4 • • • v 2 v 1 in G. Using the Z kantimagic labeling (k ≥ n) found in Theorem 2.1, we label P n .Label the edges of C G (n) in the following way: 2 → v 1 v 3 , −2 → v 3 v 5 , 2 → v 5 v 7 , ..., 2 → v n−3 v n−1 , −2 → v n−1 v n , 2 → v n v n−2 , −2 → v n−2 v n−4 , ..., 2 → v 4 v 2 and −2 → v 2 v 1 .This is a magic labeling of C G (n) with magic-value 0. Now, overlay the labelings of P n and C G (n), by identifying the vertices and edges (and adding their values).All of the vertex labels of G are distinct (mod k).We need to check if edges e 1 and e n−1 are still non-zero (mod k).Edge e i was initially labeled i+1 2 , if i is odd.Adding −2 to i+1 2 yields a non-zero value (mod k), for all k ≥ n = 4m.Thus, P 2 n is Z k -antimagic for all k ≥ n.Case 4. n = 4m + 2:Let k ≥ n + 1.Using the Z k -antimagic labeling found in Theorem 2.1, we label P n .We label the edges of C G (n) in the following way:2 → v 1 v 3 , −2 → v 3 v 5 , 2 → v 5 v 7 , . .., −2 → v n−3 v n−1 , 2 → v n−1 v n , −2 → v n v n−2 , 2 → v n−2 v n−4 , . .., 2 → v 4 v 2 and −2 → v 2 v 1 .This is a magic labeling of C G (n) with magic-value 0. Now, overlay the labelings of P n and C G (n), by identifying the vertices and edges (and adding their values).All of the vertex labels of G are distinct (mod k).We need to check if edges e 1 and e n−1 are still non-zero (mod k).Edge e i was initially labeled i+1 2 , if is odd.In G, edge e 1 is labeled 1+1 2 − 2 and edge e n−1 is labeled n−1+1 2 + 2, which are both non-zero (mod k), for all k ≥ n + 1.Thus, P 2 n is Z k -antimagic for all k ≥ n + 1.
1, we label P n .We label all of the edges of C G (n) with 1, which gives it a magic labeling with magic-value 2. Now, overlay the labelings of P n and C G (n), by identifying the vertices and edges (and adding their values).All of the vertex labels of G are distinct (mod k).We need to check if edges e 1 and e n−1 are still non-zero (mod k).Edge e i was initially labeled with one of the following: i+3 2 , if i is odd and 1 ≤ i ≤ 2m − 3; otherwise k -antimagic for all k ≥ n.Case 2. n = 4m + 3: Let k ≥ n.