Properly even harmonious labeling of a union of stars

A function f is defined as an even harmonious labeling on a graph G with q edges if f: V (G) →0, 1, …, 2 q is an injection and the induced function f *: E (G) →0, 2, …, 2 (q − 1) defined by f * (u v) = f (u) + f (v) (m o d 2 q) is bijective. A properly even harmonious labeling is an even harmonious labeling in which the codomain of f is 0, 1, …, 2 q − 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 1, and n ≥ 2. We also conclude that S n 1 ∪ S n 2 ∪ … ∪ S n k is properly even harmonious when k ≥ 2, n i ≥ 2 for all i and give some additional results on combinations of star and banana graphs.