Super H -Antimagic Total Covering for Generalized Antiprism and Toroidal Octagonal Map

Let G be a graph and H ⊆ G be subgraph of G . The graph G is said to be ( a,d ) - H antimagic total graph if there exists a bijective function f : V ( H ) ∪ E ( H ) ⟶ 1 , 2 , 3 , .. . , | V ( H )| + | E ( H )| { } such that, for all subgraphs isomorphic to H , the total H weights W ( H ) � W ( H ) � 􏽐 x ∈ V ( H ) f ( x ) + 􏽐 y ∈ E ( H ) f ( y ) forms an arithmetic sequence a,a + d, a + 2 d, . .. , a + ( n − 1 ) d , where a and d are positive integers and n is the number of subgraphs isomorphic to H . An ( a,d ) - H antimagic total labeling f is said to be super if the vertex labels are from the set 1 , 2 , . .. , | V ( G ) { } . In this paper, we discuss super ( a, d ) - C 3 -antimagic total labeling for generalized antiprism and a super ( a,d ) - C 8 -antimagic total labeling


Introduction
All the graphs that we consider in this works are finite, simple, and connected. Let G be a graph with vertex set and edge set denoted by V(G) and E(G), respectively. For the cardinality of vertex set and edge set, we use the notation |V(G)| and |E(G)|, respectively. For basic definitions and terminology related to graph theory, the readers can see the book by Gross et al. [1].
A graph labeling is a map f that sends some of the graph elements (vertices or edges or both) to the set of positive integers. If the domain set of f is the set of vertices (edges), then f is called vertex (edge) labeling. If the domain set is V(G) ∪ E(G), then f is called total labeling. Let G be a graph and H 1 , H 2 , . . . , H k be subgraphs of G. We say that the graph G has an H 1   Kotzig and Rosa [2] and Enomoto et al. [3] introduced the concept of edge-magic and super edge-magic labeling. Gutierrez and Llado [4] first studied the H (super) magic coverings of a graph G. ey proved that the cycle C n and path P n are P m super magic for some m. e cycle (super) magic behavior of some classes of connected graphs is studied in Llado et al. [5]. ey proved that prisms, windmills, wheels, and books are C m -magic for some m.
Maryati et al. [6] 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 m -supermagic for some c and m. Maryati et al. [7] and Salman et al. [8] proved that certain families of trees are path-supermagic. Ngurah et al. [9] proved that triangles, chains, ladders, wheels, and grids are cycle-supermagic. Inaya et al. [10] firstly introduced the concept of H-magic decomposition and H-antimagic decomposition. ey showed that, for any graceful tree T with n edges, the complete graph K 2n+1 admits (a, d) − T antimagic decomposition for some a and all even differences 0 ≤ d ≤ n + 1.
ey also proved that if any tree T with n edges admits α labeling, then the complete bipartite graph K n,n admits an (a, d) − T antimagic decomposition for some a and d having same parity as n. e condition on the existence of C 2k super magic decomposition of complete n partite graph and its copies were given by Lian [11].
e H-supermagic decomposition of antiprisms is described by Hendy in [12] and the H-supermagic decompositions of the lexicographic product of graphs are discussed by Hendy et al. in [13]. In [14], Hendy et al. examined the existence of super (a, d) − H magic labeling for toroidal grids and toroidal triangulations. Recently, Fenovcikova et al. [15] proved that wheels are cycle antimagic.
In this paper, we discuss the Super (a, d)-C 3 -antimagic total labeling for generalized antiprism and a Super (a, d)-C 8 -antimagic total labeling for toroidal octagonal map. We proved that the generalized antiprism A s r admits (a, d)-C 3 -antimagic total labeling for d � 0, 1 and the toroidal octagonal map O r s admits a Super (a, d)-C 8 -antimagic total labeling, for d � 1, 2, . . . , 7.

Results on Super (a, d)-C 3 -Antimagic Total
Covering of Generalized Antiprism A s r An r-sided generalized antiprism A s r is defined as a polyhedron which is composed of s parallel copies of some particular r-sided polygon and connected by an alternating band of triangles. Figure 1 represents the labeled graph of generalized antiprism A s r . We denote its vertex set and edge set by V(A s r ) and E(A s r ), respectively. e vertex set and the edge set of the generalized antiprism A s r can be defined as follows: (3) It is easy to observe that |V(A s r )| � rs and |E(A s r )| � 3rs − 2r. We first give an upper bound for d such that A s r admits a super (a, d)-C 3 -antimagic covering. 3 -antimagic covering and a 3 , a 3 + d, a 3 + 2d, . . . , a 3 + (2rs − 2r − 1)d} be the set of C 3 weights. e minimum weight on cycle C 3 is at least 12 + 3rs which is the sum of the smallest vertex labels (1, 2, 3) and sum of smallest edge labels (rs + 1, rs + 2, rs + 3). us, On the contrary, the maximum possible C 3 -weight is the sum of three largest possible vertex labels, namely, rs − 2, rs − 1, rs, and three the largest possible edge labels from the set, From (4) and (5), an upper bound for the parameter d can be obtained as us, we have arrived at the desired result.
} be a total labeling of generalized antiprism A s r defined as follows: Under the labeling ϕ, the weights of 3-cycles z j i are 2 Journal of Mathematics And, the weights of 3-cycles f For j � odd, the label on vertices x j i is defined as For j � even, the label on edges (x For j � odd, the label on edges (x e label on edges (x And, the label on edges (x Under the labeling χ, the weights of 3-cycle z (21) Observe that the weights W(z j i ) and W(f j i ) form an arithmetic progression with common difference 2 starting from 7rs + 4, 7rs + 6 and ending at 11rs − 4r + 2. is implies that the defined labeling is a super (7rs + 4, 2)-C 3 -antimagic total covering.