Cyclic Super Magic Labelings for Toroidal and Klein-Bottle Fullerenes

Abstract

In the present paper, we investigate the cyclic-supermagic behavior of toroidal and Klein-bottle graph.INDEX TERMS Toroidal fullerene, Klein-bottle fullerene, toroidal polyhex, Klein-bottle polyhex, cyclic supermagic labeling. I. INTRODUCTION AND DEFINITIONSFullerenes, the third type of carbon, have ended up essential atoms in science and innovation. In [1], [6], a massive literature is discussed about fullerenes along with their applications in science and innovation. Fullerenes are atoms made altogether out of carbon that were found in 1985 at Rice University. A large number of the viable uses of fullerenes take after straightforwardly from their uncommon properties. Significant chemical, physical and optical properties have been discussed in [7], which makes fullerenes key segments for the eventual fate of nano-electromechanical frameworks. The most significant properties are displayed as its high electron affinity and oxidation of the atom. Fullerenes are amazingly solid atoms, ready to oppose extraordinary weights. In optical properties fullerenes have indicated specific guarantees in optical restricting and power subordinate refractive list. The uses of fullerene are utilizations in solar cell, hydrogen gas storage devices, harden metals and alloys, interdigitated capacitors (IDCs), treatment of AIDS and also in magnetic resonance imaging (MRI).Let G = (V (G), E(G)) be a finite simple graph, where V (G) denote vertex set and E(G) denote edge set of G. The cardinality of vertices is denoted by |V (G)| and cardinality of edges denoted by |E(G)| respectively.The associate editor coordinating the review of this manuscript and approving it for publication was Luca Barletta.Let K 1 , K 2 , . . . , K t be a collection of subgraphs of G then G has (K 1 , K 2 , . . . , K t )-edge-covering, if every edge of E(G) belongs to a subgraph K i , 1 ≤ i ≤ t. If each subgraph K i is isomorphic to some graph K , then G admits K -covering.Consider G has K -covering. A bijective function f from the setThe weight of a subgraph K , of G under f is the sum of vertices labels and edges labels correspond to K , . The total labeling f is a K -magic labeling of G if the weight of each subgraph isomorphic to K is equal to a given constant C. The graph G is K -magic if G has K -magic labeling. Moreover, if we used smallest possible labels for vertices of G, that is if f (v) ∈ {1, 2, 3, . . . , |V (G)|}, then G is said to be H -supermagic.In 2005 Gutierrez and Llado [18] first introduced the idea of H -supermagic labeling. In [18] they showed that for some values of n the complete bipartite graph K n,m are K 1,n -supermagic. They also showed that for some n the path P n and the cycle C n are P h -supermagic. Later in 2007 Llado and Moragas [23] described some C n -super-magic graphs. Jeyanthi and Selvagopal [27], [29] discussed the H -super magic strength of graphs and some C 4 -super magic graphs moreover some results of C 3 -super magic graphs can be found in [28]. In [24] the author showed that if a graph is C-supermagic then disjoint ...