Properly even harmonious labeling of a union of stars - Henderson, Zachary M. (Electronic Journal of Graph Theory and Applications (EJGTA), 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?" We conclude that kSn is properly even harmonious if and only if k is even or k is odd, k > 1, and n ≥ 2. We also conclude that Sn1 ∪ Sn2 ∪ … ∪ Snk is properly even harmonious when k ≥ 2, ni ≥ 2 for all i and give some additional results on combinations of star and banana graphs.