Zero-sum $K_m$ over $\mathbb{Z}$ and the story of $K_4$

Abstract

We prove the following results solving a problem raised in [Y. Caro, R. Yuster, On zero-sum and almost zero-sum subgraphs over $\mathbb{Z}$, Graphs Combin. 32 (2016), 49--63]. For a positive integer $m\ge 2$, $m\ne 4$, there are infinitely many values of $n$ such that the following holds: There is a weighting function $f:E(K_n)\to \{-1,1\}$ (and hence a weighting function $f:E(K_n)\to \{-1,0,1\}$), such that ${\sum }_{e\in E(K_n)}f(e)=0$ but, for every copy $H$ of $K_m$ in $K_n$, ${\sum }_{e\in E(H)}f(e)\ne 0$. On the other hand, for every integer $n\ge 5$ and every weighting function $f:E(K_n)\to \{-1,1\}$ such that $|{\sum }_{e\in E(K_n)}f(e)|\le \left(\genfrac{}{}{0ex}{}{n}{2}\right)-h(n)$, where $h(n)=2(n+1)$ if $n\equiv 0$ (mod $4$) and $h(n)=2n$ if $n\not\equiv 0$ (mod $4$), there is always a copy $H$ of $K_4$ in $K_n$ for which ${\sum }_{e\in E(H)}f(e)=0$, and the value of $h(n)$ is sharp.