On Zero-Sum Spanning Trees and Zero-Sum Connectivity

We consider $2$-colourings $f : E(G) \rightarrow \{ -1 ,1 \}$ of the edges a graph $G$ with colours $-1$ and $1$ in $\mathbb{Z}$. A subgraph $H$ is said to be zero-sum under $f$ if $f(H) := \sum_{e\in E(H)} f(e) =0$. study following type questions, several cases obtaining best possible results: Under which conditions on $|f(G)|$ can we guarantee existence spanning tree $G$? The types are complete graphs, $K_3$-free $d$-trees, maximal planar graphs. also answer question when any such colouring contains path or diameter at most $3$, showing passing that diameter-$3$ condition possible. Finally, give, for $G = K_n$, sharp bound $|f(K_n)|$ by an interesting connectivity property forced, namely two vertices joined length $4$.
 One feature this paper proof Interpolation Lemma leading Master Theorem from many above results follow independent interest.