Properly even harmonious labeling of a union of stars - Henderson, Zachary M. (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2024)
A function f is defined as an even harmonious labeling on a graph G with q edges if f : V(G)→0, 1, …, 2q is an injection and the induced function f* : E(G)→0, 2, …, 2(q − 1) defined by f*(uv)=f(u)+f(v) (mod2q) is bijective. A properly even harmonious labeling is an even harmonious labeling in which the codomain of f is 0, 1, …, 2q − 1, and a strongly harmonious labeling is an even harmonious labeling that also satisfies the additional condition that for any two adjacent vertices with labels u and v, 0 < u + v ≤ 2q. In , Gallian and Schoenhard proved that Sn1 ∪ Sn2 ∪ … ∪ Snt is strongly even harmonious for n1 ≥ n2 ≥ … ≥ nt and t < n1/2 + 2. In this paper, we begin with the related question “When is the graph of k n-star components, G = kSn, properly even harmonious?