Super (a, d)-H-antimagic labeling of subdivided graphs

A simple graph G = (V , E) admits an H-covering, if every edge in E(G) belongs to a subgraph of G isomorphic to H. A graph G admitting an H-covering is called an (a, d)-H-antimagic if there exists a bijective function f ∶ V(G) ∪ E(G) → {1, 2, . . . , ∣V(G)∣ + ∣E(G)∣} such that for all subgraphs H′ isomorphic to H the sums∑v∈V(H′) f(v) +∑e∈E(H′) f(e) form an arithmetic sequence {a, a + d, . . . , a + (t − 1)d}, where a > 0 and d ≥ 0 are integers and t is the number of all subgraphs of G isomorphic to H. Moreover, if the vertices are labeled with numbers 1, 2, . . . , ∣V(G)∣ the graph is called super. In this paper we deal with super cycle-antimagicness of subdivided graphs. We also prove that the subdivided wheel admits an (a, d)-cycleantimagic labeling for some d.


Introduction
The H-(super)magic labelings were rst studied by Gutiérrez and Lladó [1] as an extension of the edgemagic and super edge-magic labelings introduced by Kotzig and Rosa [2] and Enomoto, Lladó, Nakamigawa and Ringel [3], respectively. In [1] are considered star-(super)magic and path-(super)magic labelings of some connected graphs and it is proved that the path P n and the cycle C n are P h -supermagic for some h. Lladó and Moragas [4] studied the cycle-(super)magic behavior of several classes of connected graphs. They proved that wheels, windmills, books and prisms are C h -magic for some h. Maryati, Salman, Baskoro, Ryan and Miller [5] and also Salman, Ngurah and Izzati [6] proved that certain families of trees are path-supermagic. Ngurah, Salman and Susilowati [7] proved that chains, wheels, triangles, ladders and grids are cycle-supermagic. Maryati, Salman and Baskoro [8] investigated the G-supermagicness of a disjoint union of c copies of a graph G and showed that the disjoint union of any paths is cP h -supermagic for some c and h.
The (a, d)-H-antimagic labeling was introduced by Inayah, Salman and Simanjuntak [9]. In [10] there are investigated the super (a, d)-H-antimagic labelings for some shackles of a connected graph H. In [11] was proved that wheels are cycle-antimagic. In [12] it was shoved that if a graph G admits a (super) (a, d)-Hantimagic labeling, where d = E(H) − V(H) , then the disjoint union of m copies of the graph G, denoted by mG, admits a (super) (b, d)-H-antimagic labeling as well. Rizvi, et al. [13] proved the disjoint union of isomorphic copies of fans, triangular ladders, ladders, wheels, and graphs obtained by joining a star K ,n with K , and also disjoint union of non-isomorphic copies of ladders and fans are cycle-supermagic.
In this paper we will discuss a super cycle-atimagicness of subdivided graphs. We show that the property to be super (a, d)-H-antimagic is hereditary according to the operation of subdivision of edges. We prove that if a graph G is super cycle-antimagic then the subdivided graph S(G) also admits a super cycle-antimagic labeling. Moreover, we show that the subdivided wheel is super (a, d)-cycle-antimagic for wide range of di erences.

Subdivided graphs
Let us consider the graph S(G) obtained by subdividing some edges of a graph G, thus by inserting some new vertices to the original graph G. Equivalently, the graph S(G) can by obtained from G by replacing some edges of G by paths. The topic of subdivided graphs has been widely studied in recent years, for example see [14].
Let G be a graph admitting H-covering given by t subgraphs H , H , . . . , H t isomorphic to H. Let us consider the subgraphs S G (H i ), i = , , . . . , t, corresponding to H i in S(G). If these subgraphs are all isomorphic to a graph, let us denote it by the symbol S G (H), then the graph S(G) admits S G (H)-covering.
The next theorem shows that the property of being super (a, d)-H-antimagic is hereditary according to the operation of subdivision of edges.

Proof. Let G be a super (a, d)-H-antimagic graph and let
such that the vertices of G are labeled with numbers , , . . . , V(G) and the weights of subgraphs form an arithmetic progression a, a + d, . . . , a + (t − )d, where a > and d ≥ are two integers, i.e., Let us consider the graph S(G) obtained from G by inserting p new vertices, say v , v , . . . , v p , to the edges of G. Let S G (H i ), i = , , . . . , t, be all subgraphs of S(G) isomorphic to S G (H). Then S(G) admits the S G (H)covering. Let r denote the number of new vertices inserted to every subgraph S G (H i ), i = , , . . . , t. We de ne a labeling g of S(G) in the following way Evidently, the vertices of S(G) are labeled with distinct numbers , , . . . , V(G) + p. Let us choose an orientation of edges in G. According to this orientation we orient the edges in S(G). To an arc uv in G there will correspond the oriented path P uv with initial vertex u and terminal vertex v in S(G). The arcs of S(G) we label such that and uw is an arc on P uv , The edges are labeled with distinct numbers from the set under the labeling g. Immediately using the structure of the subgraph S G (H i ) and the de nition of the labeling g we get As E(H i ) = E(H) for i = , , . . . , t we obtain that the weights of S G (H i ) depend on the weights of H i which form an arithmetic sequence with a di erence d, see (1). This implies that the set of weights S G (H i ) also forms an arithmetic sequences with the di erence d and the initial term This concludes the proof.
Combining Theorem 2.1 with some results on (a, d)-cycle-antimagic graphs we immediately obtain new classes of graphs that are (b, d)-cycle-antimagic. Note, that it is not needed to consider only regular subdivisions of graphs.

Subdivided wheels
A wheel W n is a graph obtained by joining a single vertex to all vertices of a cycle on n vertices. The vertex of degree n is called the central vertex, or the hub vertex, and the remaining vertices are called the rim vertices.
The edges adjacent to the central vertex are called spokes and the remaining edges are called rim edges. Let us denote by the symbol W n (r, s) the graph obtained by inserting r, r ≥ , new vertices to every rim edge and s, s ≥ , new vertices to every spoke in the wheel W n . Note, that the graph isomorphic to subdivided wheel W n (r, ) is also known as the Jahangir graph J n,r+ . In [11] it was proved that wheels are cycle-antimagic. Immediately using Theorem 2.1 we obtain that subdivided wheels admit cycle-antimagic labeling with di erence 1. In the next theorem we will deal with the cycle-antimagicness of the subdivided wheel W n ( , ). We prove that this graph admits a super (a, d)-C -antimagic labeling for d ∈ { , , . . . , }. Proof. Let us denote the vertices and edges of W n ( , ) such that where the indices are taken modulo n.
We denote by the symbol C i , ≤ i ≤ n, the -cycle such that C i = cw i u i v i u i+ w i+ , where the index i is taken modulo n. Under the labeling g d , the weights of C i are as follows.
It is a simple mathematical exercise to prove that for every i, ≤ i ≤ n, the -cycle-weights are: wt g (C i ) = n + , for ≤ i ≤ n, Hence the weights of cycles C form an arithmetic sequence with di erences d = , , , , , respectively. This concludes the proof.
Combining Theorem 2.1 and Theorem 3.3 we immediately obtain the following result. In the next section we will deal with the subdivided wheel W n (r, ), n ≥ , r ≥ . Let us denote the vertices and the edges of W n (r, ) such that The subdivided wheel W n (r, ), n ≥ , r ≥ , has n vertices of degree 3, nr vertices of degree 2 and one vertex of degree n. The size of W n (r, ) is n(r + ).
The subdivided wheel W n (r, ) admits the C r+ -covering consisting of n cycles C r+ . Let us denote these cycles by the symbols C i r+ , i = , , . . . , n, such that C i r+ = cv i u i u i . . . u i r v i+ . The following theorem shows the existence of a super (a, d)-C r+ -antimagic labeling for W n (r, ) for every odd di erence form 1 up to r − .
Under the labeling f d the weights of cycles C i r+ , i = , , . . . , n − , are as follows.
In the next theorem we prove that the graph W n (r, ) admits super (a, d)-C r+ -antimagic labelings also for even di erences.
The labeling g d is a bijection. Under the labeling g d the weights of cycles C i r+ , i = , , . . . , n − , are the following. For the weight of the cycle C n r+ we obtain: g d (e) = g d (c) + g d (v n ) + g d (v ) + r j= g d (u n j ) + g d (cv n ) + g d (cv ) + g d (v n u n ) + r− j= g d (u n j u n j+ ) + g d (u n r v ) = + n + + + r j= jn + + n + nr + n + + nr + n + + nr + n + + +d j= nr + n + − jn + r− j= +d nr + n + − jn + nr + n + = nr + nr + n + r + + nd − d .
We showed that g d is a super (a, d)-C r+ -antimagic labeling of W n (r, ) for d ≡ (mod ), ≤ d ≤ r − and a = nr + nr + n + r + − dn + d .
Combining Theorem 3.5 and Theorem 3.6 we immediately obtain that the subdivided wheel W n (r, ), n ≥ , is cycle-antimagic for wide range of di erences.
Moreover, using Theorem 2.1, we can extend this result also for subdivided wheels in which not only rim edges but also spokes are subdivided.