Discrete Mathematics ( IF 0.770 ) Pub Date : 2020-08-01 , DOI: 10.1016/j.disc.2020.112074
S. Akbari; H.R. Maimani; Leila Parsaei Majd; I.M. Wanless

Given a $t$-$\left(v,k,\lambda \right)$ design, $\mathcal{D}=\left(X,\mathcal{B}\right)$, a zero-sum $n$-flow of $\mathcal{D}$ is a map $f:\mathcal{B}⟶\left\{±1,\dots ,±\left(n-1\right)\right\}$ 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 $\mathrm{STS}\left(uw\right)$ and for an $\mathrm{STS}\left(2v+7\right)$ for $v\equiv 1\phantom{\rule{1em}{0ex}}\left(\mathrm{mod}\phantom{\rule{1em}{0ex}}4\right)$, if there are $\mathrm{STS}\left(u\right)$, $\mathrm{STS}\left(w\right)$ and $\mathrm{STS}\left(v\right)$ such that the $\mathrm{STS}\left(u\right)$ and $\mathrm{STS}\left(v\right)$ both have a zero-sum $k$-flow. In 2015, it was conjectured that for $v>7$ every $\mathrm{STS}\left(v\right)$ admits a zero-sum 3-flow. Here, it is shown that many cyclic $\mathrm{STS}\left(v\right)$ have a zero-sum 3-flow. Also, we investigate the existence of zero-sum flows for some Steiner quadruple systems.

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