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

TL;DR

This paper investigates zero-sum weightings on complete graphs over integers, proving existence and non-existence results for certain configurations, especially highlighting the special case of $K_4$, and establishing sharp bounds.

Contribution

It provides new results on zero-sum weightings over $ extbf{Z}$ for complete graphs, especially addressing the case of $K_4$, which was previously unresolved.

Findings

Existence of infinitely many $n$ with zero-sum weightings avoiding zero-sum $K_m$ for $m eq 4$

For all $n eq 4$, specific bounds guarantee the presence of zero-sum $K_4$

The bounds on the sum deviation are sharp for the existence of zero-sum $K_4$

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 $m\geq 2$, $m\neq 4$, 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 $K_n$, $\sum_{e\in E(H)}f(e)\neq 0$. On the other hand, for every integer $n\geq 5$ and every weighting function $f:E(K_n)\to \{-1,1\}$ such that $|\sum_{e\in E(K_n)}f(e)|\leq \binom{n}{2}-h(n)$, 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.