Dekomposisi H-Super Anti Ajaib Atas Graf C_n >_o S_n - Aditya Putra Pratama; Winarni; Tiara Uni Raudyna (Limits: Journal of Mathematics and Its Applications, 2024)
The concept of an H-Magic decomposition of a graph G is formed based on the concept of decomposition and the concept of labeling a graph. The set A = {H1, H2, ..., Hk} is a set of subgraphs of graph G which is called a decomposition of G if the union of all Hi (for 1 less than or equal to i less than or equal to k) equals G, and the intersection of edge sets E(Hi) and E(Hj) is empty for all i not equal to j. If every subgraph Hi resulting from the decomposition of graph G is isomorphic to a subgraph H of G, then the set {H1, H2, ..., Hk} is called an H-decomposition of G. Graph G is said to have an H-Magic decomposition if there is a bijective mapping from the set of vertices and edges of G, that is V(G) union E(G), to the set of integers from 1 to (number of vertices of G plus number of edges of G), such that the total sum of labels on the vertices and edges in each subgraph Hi is constant. If the total label sum of vertices and edges in each subgraph Hi forms an arithmetic sequence where the difference between each subgraph’s total weight is one, then graph G is said to have an H-Anti Magic decomposition. In this study, the H-Super Anti Magic decomposition of the graph C_n lexicographic product with S_n (denoted as C_n o S_n) is investigated. First, the properties of the graph C_n o S_n are explored along with the chosen subgraphs. Then, based on the selected subgraph, a labeling pattern is constructed on the graph C_n o S_n such that the total weight of each subgraph forms an arithmetic sequence with a difference of one. From the labeling pattern, a bijective labeling function is created using an arithmetic sequence approach. Based on this labeling function, it is shown that the subgraphs of C_n o S_n form an H-decomposition of C_n o S_n. The final result of this research is that the graph C_n o S_n admits an H-Super Anti Magic decomposition, where the magic weight of each subgraph Hi is: w_n(Hi) = 2n cubed plus 4n squared plus 3n plus 2 plus i, for 1 less than or equal to i less than n w_n(Hi) = 2n cubed plus 4n squared plus 3n plus 2, for i equal to n with the condition that n is greater than or equal to 3, and n is a natural number.