Weighted-1-antimagic graphs of prime power order

Suppose G is a graph, k is a non-negative integer. We say G is weighted- k -antimagic if for any vertex weight function w : V → N , there is an injection f : E → { 1 , 2 , … , ∣ E ∣ + k } such that for any two distinct vertices u and v , ∑ e ∈ E ( v ) f ( e ) + w ( v ) ≠ ∑ e ∈ E ( u ) f ( e ) + w ( u ) . There are connected graphs G ≠ K 2 which are not weighted-1-antimagic. It was asked in Wong and Zhu (in press) [13] whether every connected graph other than K 2 is weighted-2-antimagic, and whether every connected graph on an odd number of vertices is weighted-1-antimagic. It was proved in Wong and Zhu (in press) [13] that if a connected graph G has a universal vertex, then G is weighted-2-antimagic, and moreover if G has an odd number of vertices, then G is weighted-1-antimagic. In this paper, by restricting to graphs of odd prime power order, we improve this result in two directions: if G has odd prime power order p z and has total domination number 2 with the degree of one vertex in the total dominating set not a multiple of p , then G is weighted-1-antimagic. If G has odd prime power order p z , p ≠ 3 and has maximum degree at least ∣ V ( G ) ∣ − 3 , then G is weighted-1-antimagic.