PELABELAN SELIMUT AJAIB SUPER PADA GRAF LINTASAN

Abstract

Let ðº = (ð‘‰, ð¸) be a simple graph. An edge covering of ðº is a family of subgraphs ð»1 , … , ð»ð‘˜ such that each edge of ð¸(ðº) belongs to at least one of the subgraphs ð»ð‘– , 1 ≤ 𑖠≤ ð‘˜. If every ð»ð‘– is isomorphic to a given graph ð», then the graph ðº admits an ð» − covering. Let ðº be a containing a covering ð», and ð‘“ the bijectif function ð‘“: (𑉠∪ ð¸) → {1,2,3, … , |ð‘‰| + |ð¸|} is said an ð» −magic labeling of ðº if for every subgraph ð» ′ = (𑉠′ ,ð¸ ′ ) of ðº isomorphic to ð», is obtained that ∑ ð‘“(ð‘‰) + ∑ ð‘“(ð¸) ð‘’∈ð¸(ð»â€² ð‘£âˆˆð‘‰(ð» ) ′ ) is constant. ðº is said to be ð» −super magic if ð‘“(ð‘‰) = {1, 2, 3, … , |ð‘‰|}. In this case, the graph ðº which can be labeled with ð»-magic is called the covering graph ð» −magic. The sum of all vertex labels and all edge labels on the covering ð» − super magic then obtained constant magic is denoted by ∑ ð‘“(ð»). The duplication graph 2 of graph ð·2 (ðº) is a graph obtained from two copies of graph ðº, called ðº and ðº ′ , with connecting each respectively vertex ð‘£ in ðº with the vertexs immediate neighboring of 𑣠′ in ðº ′ . The purpose of this study is to obtain a covering super magic labeling for of ð·2 (ð‘ƒð‘š) on (ð·2 (ð‘ƒð‘› )) for 𑛠≥ 4 and 3 ≤ 𑚠≤ 𑛠− 1. In this paper, we have showed that duplication path graph (ð·2 (ð‘ƒð‘› )) has ð·2 (ð‘ƒð‘š) covering super magic labeling for 𑛠≥ 4 and 3 ≤ 𑚠≤ 𑛠− 1 with constant magic for all covering is ∑ ð‘“(ð·2 (ð‘ƒð‘š) (ð‘ ) ) = ∑ ð‘“(ð·2 (ð‘ƒð‘š) (ð‘ +1) )