Academic literature on the topic 'Super Magic Labeling'

Abstract An edge-magic total labeling of an (n,m)-graph G = (V,E) is a one to one map λ from V(G) ∪ E(G) onto the integers {1,2,…,n + m} with the property that there exists an integer constant c such that λ(x) + λ(y) + λ(xy) = c for any xy ∈ E(G). It is called super edge-magic total labeling if λ (V(G)) = {1,2,…,n}. Furthermore, if G has no super edge-magic total labeling, then the minimum number of vertices added to G to have a super edge-magic total labeling, called super edge-magic deficiency of a graph G, is denoted by μs(G) [4]. If such vertices do not exist, then deficiency of G will be + ∞. In this paper we study the super edge-magic total labeling and deficiency of forests comprising of combs, 2-sided generalized combs and bistar. The evidence provided by these facts supports the conjecture proposed by Figueroa-Centeno, Ichishima and Muntaner-Bartle [2].

AbstractA super edge-magic total (SEMT) labeling of a graph ℘(V, E) is a one-one map ϒ from V(℘)∪E(℘) onto {1, 2,…,|V (℘)∪E(℘) |} such that ∃ a constant “a” satisfying ϒ(υ) + ϒ(υν) + ϒ(ν) = a, for each edge υν ∈E(℘), moreover all vertices must receive the smallest labels. The super edge-magic total (SEMT) strength, sm(℘), of a graph ℘ is the minimum of all magic constants a(ϒ), where the minimum runs over all the SEMT labelings of ℘. This minimum is defined only if the graph has at least one such SEMT labeling. Furthermore, the super edge-magic total (SEMT) deficiency for a graph ℘, signified as $\mu_{s}(\wp)$ is the least non-negative integer n so that ℘∪nK1 has a SEMT labeling or +∞ if such n does not exist. In this paper, we will formulate the results on SEMT labeling and deficiency of fork, H -tree and disjoint union of fork with star, bistar and path. Moreover, we will evaluate the SEMT strength for trees.

Let G be a finite, simple, and undirected graph with vertex set V G and edge set E G . A super edge-magic labeling of G is a bijection f : V G ∪ E G ⟶ 1,2 , … , V G + E G such that f V G = 1,2 , … , V G and f u + f u v + f v is a constant for every edge u v ∈ E G . The super edge-magic labeling f of G is called consecutively super edge-magic if G is a bipartite graph with partite sets A and B such that f A = 1,2 , … , A and f B = A + 1 , A + 2 , … , V G . A graph that admits (consecutively) super edge-magic labeling is called a (consecutively) super edge-magic graph. The super edge-magic deficiency of G , denoted by μ s G , is either the minimum nonnegative integer n such that G ∪ n K 1 is super edge-magic or + ∞ if there exists no such n . The consecutively super edge-magic deficiency of a graph G is defined by a similar way. In this paper, we investigate the (consecutively) super edge-magic deficiency of subdivision of double stars. We show that, some of them have zero (consecutively) super edge-magic deficiency.