Zero-sum flows for Steiner systems
Given a t-(v, k, λ) design, D=(X,B), a zero-sum n-flow of D is
a map f:B→{±1,...,±(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(uw)
and for an STS(2v+7) for v=1 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.