Zero‐sum flows in designs

Let D be a t ‐ ( v, k , λ) design and let Ni(D) , for 1 ≤ i ≤ t , be the higher incidence matrix of D , a ( 0 , 1 )‐matrix of size , where b is the number of blocks of D . A zero‐sum flow of D is a nowhere‐zero real vector in the null space of N1 ( D ). A zero‐sum k‐flow of D is a zero‐sum flow with values in { 1 , …, ±( k − 1 )}. In this article, we show that every non‐symmetric design admits an integral zero‐sum flow, and consequently we conjecture that every non‐symmetric design admits a zero‐sum 5‐flow. Similarly, the definition of zero‐sum flow can be extended to Ni ( D ), 1 ≤ i ≤ t . Let be the complete design. We conjecture that Nt ( D ) admits a zero‐sum 3‐flow and prove this conjecture for t = 2 . © 2011 Wiley Periodicals, Inc. J Combin Designs 19:355‐364, 2011