Zero-sum flows for Steiner systems

Abstract: Given a $t$-$(v, k, \lambda)$ design, $\mathcal{D}=(X,\mathcal{B})$, a zero-sum $n$-flow of $\mathcal{D}$ is a map $$f : \mathcal{B}\longrightarrow {\pm1,\ldots, \pm(n-1)}$$ such that for any point $x\in X$, the sum of $f$ over all blocks incident with $x$ is zero. For a positive integer $k$, we find a zero-sum $k$-flow for an STS$(u w)$ and for an STS$(2v+7)$ for $v\equiv 1~(\mathrm{mod}~4)$, if there are STS$(u)$, STS$(w)$ and STS$(v)$ such that the STS$(u)$ and STS$(v)$ both have a zero-sum $k$-flow. In 2015, it was conjectured that for $v>7$ every STS$(v)$ admits a zero-sum $3$-flow. Here, it is shown that many cyclic STS$(v)$ have a zero-sum $3$-flow. Also, we investigate the existence of zero-sum flows for some Steiner quadruple systems.