Pelabelan Odd-Graceful pada Graf Kincir Angin Double Quadrilateral dan Gabungan Dua Graf Kincir Angin Double Quadrilateral

Abstract

Graph labeling is a mapping that pairs graph elements to positive integers with certain conditions called labels. One of the graph labeling that is known until now is graceful labeling. A graceful labeling on a graph 𝐺 with 𝑞 edges is an injective mapping 𝑓: 𝑉(𝐺) → {0,1,2,..., 𝑞} such that the edge label obtained is {0,1,2,..., 𝑞}. In 1991, Gnanajothi extended the concept of graceful labeling by introducing odd-graceful labeling. An odd-graceful labeling on a graph 𝐺 with 𝑞 edges is an injective mapping 𝑓:𝑉(𝐺) → {0,1,2,...,2𝑞 − 1} such that each edge 𝑥𝑦 is labeled with |𝑓(𝑥) – 𝑓(𝑦)| and the side labels are distinct. Thus, the edge labels of odd graceful labeling satisfy the bijective function of 𝑓∗: 𝐸(𝐺) → {1,3,5,…,2𝑞 − 1}. The purpose of this study is to analyze whether the double quadrilateral windmill graph 𝐷𝑄(𝑘) and the union of two double quadrilateral windmill graphs 2𝐷𝑄(𝑘) are odd-graceful graphs or not. The methods used in this research are pattern detection method and axiomatic deductive method. The pattern detection method is a method used to find a pattern that can help formulate a labeling pattern by applying the labeling to the graph with the smallest number of nodes. The axiomatic deductive method is a method that uses deductive proof principles that apply in mathematical logic by deriving existing theorems and then applying them to odd graceful labeling. In this study, it is found that the double quadrilateral windmill graph 𝐷𝑄(𝑘)and the union of two double quadrilateral windmill graphs 2𝐷𝑄(𝑘) are odd-graceful graphs.