On Zero-Sum and Almost Zero-Sum Subgraphs Over ℤ

For a graph H with at most n vertices and a weighing of the edges of Kn with integers, we seek a copy of H in Kn whose weight is minimal, possibly even zero. Of a particular interest are the cases where H is a spanning subgraph (or an almost spanning subgraph) and the case where H is a fixed graph. In particular, we show that relatively balanced weighings of Kn with {−r, . . . , r} guarantee almost zero-sum copies of spanning graphs with small maximum degree, guarantee zero-sum almost H-factors, and guarantee zero-sum copies of certain fixed graphs.