A C3 Magic Decomposition on Friendship Graph with Odd Order

Abstract

Let G = (V,E) is graph with a non-empty set V containing vertices and a set of edges E. Also note that if H = {H_i⊆G_i = 1,2,3,...,n} is a collection of subgraphs from G with H_i≅Hj,i ≠ j. If Hi ∩ Hj = ∅ and ⋃n(i-1)Hi = G, then graph G admits a decomposition H. Furthermore, if there are f(v) and g(e) which are vertices and edges labeling at G, the total weight of each subgraph H_i,i = 1,2,3,…,n has the same value, namely ∑_(v∈V(H_i))▒〖f(v)〗+∑_(e∈E(H_i))▒〖g(e)〗= w, then the graph G contains the magic H_i decomposition with w as the magic constant. This research shows that the friendship graph F_n with n = 2k + 1 for k∈N admits a magic -(a,d)-C_3 decomposition with a magic constant w of 29dk + 6a + 15d.