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## Preprints Archive: Abstract of IC2010060 (2010)

### Nowhere-zero flows in designs

Document info: Pages 10, Figures 0.

Let $D$ be a $t$-$(v, k, \lambda)$ design and let $N_i(D)$, for $1 \leq i \leq t$, be the higher incidence matrix of $D$, a $(0,1)$-matrix of size $\binom{v}{i}\times b$, 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 $N_1(D)$. A zero-sum k-flow of $D$ is a zero-sum flow with values in $\{\pm 1, \ldots ,\pm (k-1)\}$. 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 $N_i(D)$, $1 \leq i \leq t$. Let $D=t$-$(v,k,\binom{v-t}{k-t})$ be the complete design. We conjecture that $N_t(D)$ admits a zero-sum 3-flow and prove this conjecture for $t=2$.

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