Yuliana Ulfa, P. Purwanto
Let G be a finite graph, without loops nor multiple edges, having q edges. A function f is called properly even harmonious labeling on G if f is an injection from V (G) to {0,1, 2, ..,2q -1} and the induced function f* from E(H) to {0, 2, 4, .., 2q - 2}, with f*(xy) = (f(x) + f(y)) (mod 2q), is a bijective. If there exists such function, then G is said to be properly even harmonious. A complete tripartite graph Kl,m,n is a graph that its vertex set can be decomposed into three disjoint sets of cardinality l, m, and n, respectively, such that no two vertices within the same set are adjacent and every vertex of one set is adjacent to every vertex in the other two sets. A coconut tree is a graph formed from a star and a path by identifying the central vertex of the star and a vertex of degree one of the path. In this paper we study the properly even harmonious labelings of Kl,m,n and a union of two coconut trees. We find that Kl,m,n and a union of two coconut trees are properly even harmonious. © 2021 Author(s).
Department of Mathematics, Universitas Negeri Malang, Jalan Semarang 5, Malang, 65145, Indonesia
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