Zero-sum flows for Steiner systems

TL;DR

This paper investigates zero-sum flows in Steiner systems, establishing new results for specific cases and providing evidence supporting a conjecture that all Steiner triple systems beyond size 7 admit zero-sum 3-flows.

Contribution

The paper introduces new constructions of zero-sum flows for certain Steiner systems and explores their existence in cyclic and quadruple systems, advancing understanding of flow properties.

Findings

Constructed zero-sum k-flows for specific Steiner systems.

Many cyclic Steiner triple systems have zero-sum 3-flows.

Explored zero-sum flows in Steiner quadruple 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.