PELABELAN SELIMUT AJAIB SUPER PADA GRAF LINTASAN - Farida, N; Sudarsana, I W; Resnawati, Resnawati (JURNAL ILMIAH MATEMATIKA DAN TERAPAN, 2018)
Let ðº = (ð‘‰, ð¸) be a simple graph. An edge covering of ðº is a family of subgraphs ð»1 , â€¦ , ð»ð‘˜ such that each edge of ð¸(ðº) belongs to at least one of the subgraphs ð»ð‘– , 1 â‰¤ ð‘– â‰¤ ð‘˜. If every ð»ð‘– is isomorphic to a given graph ð», then the graph ðº admits an ð» âˆ’ covering. Let ðº be a containing a covering ð», and ð‘“ the bijectif function ð‘“: (ð‘‰ âˆª ð¸) â†’ {1,2,3, â€¦ , |ð‘‰| + |ð¸|} is said an ð» âˆ’magic labeling of ðº if for every subgraph ð» â€² = (ð‘‰ â€² ,ð¸ â€² ) of ðº isomorphic to ð», is obtained that âˆ‘ ð‘“(ð‘‰) + âˆ‘ ð‘“(ð¸) ð‘’âˆˆð¸(ð»â€² ð‘£âˆˆð‘‰(ð» ) â€² ) is constant. ðº is said to be ð» âˆ’super magic if ð‘“(ð‘‰) = {1, 2, 3, â€¦ , |ð‘‰|}. In this case, the graph ðº which can be labeled with ð»-magic is called the covering graph ð» âˆ’magic. The sum of all vertex labels and all edge labels on the covering ð» âˆ’ super magic then obtained constant magic is denoted by âˆ‘ 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 â€² , with connecting each respectively vertex ð‘£ in ðº with the vertexs immediate neighboring of ð‘£ â€² in ðº â€² . 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 ð‘› â‰¥ 4 and 3 â‰¤ ð‘š â‰¤ ð‘› âˆ’ 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 ð‘› â‰¥ 4 and 3 â‰¤ ð‘š â‰¤ ð‘› âˆ’ 1 with constant magic for all covering is âˆ‘ ð‘“(ð·2 (ð‘ƒð‘š) (ð‘ ) ) = âˆ‘ ð‘“(ð·2 (ð‘ƒð‘š) (ð‘ +1) )