Weighted-1-antimagic graphs of prime power order

Suppose GG is a graph, kk is a non-negative integer. We say GG is weighted-kk-antimagic if for any vertex weight function w:V→Nw:V→N, there is an injection f:E→{1,2,…,∣E∣+k}f:E→{1,2,…,∣E∣+k} such that for any two distinct vertices uu and vv, ∑e∈E(v)f(e)+w(v)≠∑e∈E(u)f(e)+w(u)∑e∈E(v)f(e)+w(v)≠∑e∈E(u)f(e)+w(u). There are connected graphs G≠K2G≠K2 which are not weighted-1-antimagic. It was asked in Wong and Zhu (in press) [13] whether every connected graph other than K2K2 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 GG has a universal vertex, then GG is weighted-2-antimagic, and moreover if GG has an odd number of vertices, then GG is weighted-1-antimagic. In this paper, by restricting to graphs of odd prime power order, we improve this result in two directions: if GG has odd prime power order pzpz and has total domination number 2 with the degree of one vertex in the total dominating set not a multiple of pp, then GG is weighted-1-antimagic. If GG has odd prime power order pzpz, p≠3p≠3 and has maximum degree at least ∣V(G)∣−3∣V(G)∣−3, then GG is weighted-1-antimagic.