For a complete graph $$K_n$$ of order n, an edge-labeling $$c:E(K_n)\rightarrow \{ -1,1\}$$ satisfying $$c(E(K_n))=0$$ , and spanning forest F we consider the problem to minimize $$|c(E(F'))|$$ over all isomorphic copies $$F'$$ in . In particular, ask under which additional conditions there is zero-sum copy, that is, copy with $$c(E(F'))=0$$ We show always $$|c(E(F'))|\le \Delta (F)+1$$ where $...

let $r$ be a commutative ring with zero-divisor set $z(r)$. the total graph of $r$, denoted by$t(gamma(r))$, is the simple (undirected) graph with vertex set  $r$ where two distinct vertices areadjacent if their sum lies in $z(r)$.this work considers minimum zero-sum $k$-flows for $t(gamma(r))$.both for $vert rvert$ even and the case when $vert rvert$ is odd and $z(g)$ is an ideal of $r$it is s...