Zero-Sum Copies of Spanning Forests in Zero-Sum Complete Graphs

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 $$\Delta (F)$$ maximum degree F. conjecture this bound can be improved (\Delta (F)-1)/2$$ verify for being star $$K_{1,n-1}$$ Under some simple necessary divisibility conditions, existence $$P_3$$ -factor, and, sufficiently large also $$P_4$$ -factor.