(Ck,Ck+1)- SUPERMAGIC LABELINGS OF CUPOLA GRAPH Cu(k,k+1,n) - Muhammad Issaac, Gardhani
Let G=(V,E) be a simple graph. Let H_1 and H_2 are two subgraphs of G. G=(V,E) admits (H_1,H_2)-covering if every edge in E belongs to a subgraph of G that isomorphic to H_1 or H_2. $G$ is called (H_1,H_2-Magic if there are two fixed positive integers k_1 and k_2 and a bijective function f: V? E? {1,2,3,...,|V|+|E|} such that ?_(v?V^')??f(v)?+?_(e?E^')??f(e)?=k_1 and ?_(v?? V?^")??f(v)+?_(e?E")??f(e)?=k_2 ? for every subgraph H'= (V',E') of G isomorphic to H_1 and for every subgraph H"=(V",E") of G isomorphic to H_2. Furthermore, the graph G is called (H_1,H_2)-supermagic if f(V)={1,2,...,|V|}. In this research we introduce a cupola graph. Let n?N. Let a and b are integers with 3? a< b. For i?[1,n], let C_(b,i) be a cycle graph of order b which V(C_(b,i) )={v_(i,j)?j?[1,b-1] }?{v_(i+1,1)} with v_(n+1,1)=v_1,1 and E(C_(b,i) )={v_(i,j) v_(i,j+1)?j?[1,b-2] }?{v_(i,b-1) v_(i+1,1),v_(i+1,1) v_(i,1)}. Let r=?(a-1)/2?.. Define cupola graph Cu(a,b,n)=(V,E) as follows. For odd a, V(Cu(a,b,n))=?_(i=1)^n?(V(C_(b,i))) and E(Cu(a,b,n))=(?_(i=1)^n?(E(C_(b,i) )) )?(?_(i=1)^n?({v_(i,b-r) ? v?_(i+1,r+1)}) . For even a, V(Cu(a,b,n))=(?_(i=1)^n?(V(C_(b,i) )) ?(?_(i=1)^n?(w_i ) ) dan E(Cu(a,b,n))=(?_(i=1)^n??(E(C_(b,i) ))?(?_(i=1)^n?({v_(i-1,b-r) w_i,w_i v_(i,r+1)}) ?. By using (k_1,k_2,?_1,?_2,l_1,l_2 )-balanced multiset partition, we investigate existance of (C_k,C_(k+1))-supermagic labeling on an n-gonal cupola Cu(k,k+1,n). We prove that Cu(k,k+1,n) is (C_k,C_(k+1))-supermagic for every n? 3 and k? 3.