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 < u + v ≤ 2 q . In , Gallian and Schoenhard proved that S n 1 ∪ S n 2 ∪ … ∪ S n t is strongly even harmonious for n 1 ≥ n 2 ≥ … ≥ n t and t < n 1 /2 + 2 . In this paper, we begin with the related question "When is the graph of k n -star components, G = k S n , properly even harmonious?" We conclude that k S n is properly even harmonious if and only if k is even or k is odd, k > 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.