Zero-sum flows for Steiner triple systems

TL;DR

This paper proves that most Steiner triple systems and related designs admit zero-sum 3-flows, provides bounds on the flow size, and shows how to embed smaller systems into larger ones with similar flow properties.

Contribution

It establishes the existence of zero-sum 3-flows for nearly all Steiner systems and related designs, and introduces recursive embedding results for these flows.

Findings

Most Steiner triple systems admit zero-sum 3-flows.

A bound of O(λ^2 v^2) on the flow size n is provided.

Small systems can be embedded into larger systems with compatible zero-sum flows.

Abstract

Given a $2$-$(v,k,\lambda)$ design, $\cal{S}=(X,\cal{B})$, a {\it zero-sum $n$-flow} of $\cal{S}$ is a map $f: \cal{B} \longrightarrow \{\pm 1, \ldots ,\pm (n-1)\}$ such that for any point $x\in X$, the sum of $f$ around all the blocks incident with $x$ is zero. It has been conjectured that every Steiner triple system, STS$(v)$, on $v$ points $(v>7)$ admits a zero-sum $3$-flow. We show that for every pair $(v,\lambda)$, for which a triple system, TS$(v,\lambda)$ exists, there exists one which has a zero-sum $3$-flow, except when $(v,\lambda)\in\{(3,1), (4,2), (6,2), (7,1)\}$ and except possibly when $v \equiv 10\pmod{12}$ and $\lambda = 2$. We also give a $O(\lambda^2v^2)$ bound on $n$ and a recursive result which shows that every STS$(v)$ with a zero-sum $3$-flow can be embedded in an STS$(2v+1)$ with a zero-sum $3$-flow if $v\equiv 3 \pmod 4$, a zero-sum $4$-flow if $v\equiv 3 \pmod 6$ and with a zero-sum $5$-flow if $v\equiv 1 \pmod 4$.