The Abdus Salam International Centre for Theoretical Physics 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 ( v i ) × b, 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 paper 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 D = t-(v, k, ( v−t k−t ) ) be the complete design. We conjecture that Nt(D) admits a zero-sum 3-flow and prove this conjecture for t = 2.