H-magic covering on Complete Bipartite Graph for Constructing Secret Sharing Schemes

Roswitha, Mania and Harjito, Bambang (2014) H-magic covering on Complete Bipartite Graph for Constructing Secret Sharing Schemes. DIKTI.

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    Abstract

    A graph G is an ordered pair of a sett ܸሺܩሻof vertices and a set of edges ܧሺܩሻ. An edge magiic total (EMT) labeling of a graph ܩis defined as bijective function : λ: V(G) ∪ E(G) →{1, 2, 3, …, |V(G)| + |E(G)|} such that there exist a positive integer ݇satisfying: λ(x) + λ(xy) + λ(y) = k …………….. (1) for each xy א G. We shall call kas the magic constant of graph G.A graph that admits an EMT is said to beedge magic total (EMT). Researchs on EMT labeling havebeen first introduced by Kotzig and Rosa ([18], [ 19]), then followed by W.D. Wallis et al. [30], Wijaya and Baskoro [31], Wijaya [32] , Berkman [4] and Ngurah and Baskoro [2] . Recently, Gutiérrez and Lladó [17], introduced a magic cover labelingand defined it as follows. An edge-covering of G is a family of different subgraphs H1, …, Hk such that any edge of Gbelongs to at least one of the subgraphs Hi , k i≤ ≤ 1 . Then it is said that G admits an -(H1, …, Hk )-(edge) covering.. If every Hi isomorphic to a given graph H, then we say that G admits an H-covering.. Suppose that G = (V,E) admits an H-covering. We say that a bijectivefunction |} | | | ,..., 2 , 1 { : E V E V f + → ∪ is an H-magic labeling of Gif there is a positive integer ݉ሺ݂ሻ which we call magic-sumsuch that for each subgraph of ) ' , ' ( ' E V H= of Gisomorphic to Hwe have '' (') () () (). def vV eE f Hfvfemf ∈∈ =+= ∑∑ In this case we say that Gis − H magic. If ( ) {1,...,| |} f VV= , or the labels on its vertices are the smallest labels then Gis H−super magicand we denote its supermagic sum by ݏሺ݂ሻ. x If Hisomorphic only to one edge then the concept coincide with the well-known magic valuation introduced by Rosa in 1966, and the supermagic case (see [2], [1]). In this case, an Hcoveringis also an H-decomposition.Figure 1. Shows an example of a 1,2 K - supermagic labeling of ܭଵ,଺ . Fig. 1. 1,2 K - supermagic labeling of ܭଵ,଺. In this research we study an enumeration labeling of 1,n K -magic covering to , for 2,3, 4. nn Kn= : 1. Complete bipartite graph 2,2 . K For complete bipartite graps 2,2 K we have 4 1,n K -magic covering as follows. . The sum of vertex labels (T) The magic sum ݉ሺ݂ሻ Vertices Labels Edge Labels a b C d e f g h 12 21 1 5 4 2 8 6 3 7 15 22 7 1 2 6 4 3 8 5 20 23 2 8 7 3 5 6 1 4 24 24 8 4 5 7 1 3 6 2 8 9 10 11 12 13 6 5 4 3 2 1 7 xi (a) (b) (c) (d) Fig. 2. ܭଵ,ଶ – magic covering on ܭଶ,ଶ . The critical sets of size-1 of ܭଵ,ଶ – magic covering on ܭଶ,ଶ are ܳଵଵ = {_, _, _, _, _, _, 7, _}, ܳଵଶ = {_, _, _, 6, _, _, _, _}, ܳଵଷ = {_, _, _, 3, _, _, _, _} and ܳଵସ = {_, _, _, _, _, _, _, 2} as Fig. 3 (a) (b) (c) (d) Fig. 3. The critical sets of size-1 of ܭଵ,ଶ – magic covering on ܭଶ,ଶ . To simplify, we write the critical sets as ܳ8-2012 period. The component of marketing mix strategy applied to Sembada Plus saving product of PD. BPR Bank Daerah Karanganyar had changed the consumer behavior in making decision and loyalty to save in Sembada saving. The recommendation was that there should be socialization and approach to the society to generate the public’s trust in PD. BPR Bank Daerah Karanganyar. In addition, PD. BPR Bank Daerah Karanganyar should take into account the interrelatedness of marketing mix strategy components consisting of 4 variables: Product, Price, Place and Promotion.

    Item Type: Article
    Subjects: Q Science > Q Science (General)
    Divisions: Lembaga Penelitian dan Pengabdian Kepada Masyarakat - LPPM
    Depositing User: Lia Primadani
    Date Deposited: 15 Apr 2014 02:07
    Last Modified: 19 Oct 2016 10:39
    URI: https://eprints.uns.ac.id/id/eprint/11040

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