On Zero-Sum and Almost Zero-Sum Subgraphs Over $${\mathbb {Z}}$$ Z

Abstract

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