论文标题
Novák对循环施泰纳三重系统的猜想及其概括
Novák's conjecture on cyclic Steiner triple systems and its generalization
论文作者
论文摘要
Novák在1974年猜想，对于任何循环施泰纳三重系统，$ v $带有$ v \ equiv 1 \ pmod {6} $的订单$ v $，始终可以从每个块轨道中选择一个块，以使所选的块是成对的脱节。我们考虑将此猜想的概括为环状$（v，k，λ）$ - 以$ 1 \leqλ\ leq k-1 $设计。叠加循环对称设计的多个副本表明，对于所有$ v $，概括都无法实现，但是我们认为，每当$ v $足够大，而与$ k $相比，它会保留。我们确认，当$ v $是Prime和$λ= 1 $的情况下，猜想的概括以及当$λ\ leq（k-1）/2 $和$ v $相比$ k $时也足够大。作为推论，我们表明，对于任何$ k \ geq 3 $，除了有限的许多复合订单$ v $外，每个环状$（v，k，1）$ - 无短轨道的设计是由$（v，k，k，1）$ - $ - 与众不同的差异差异。
Novák conjectured in 1974 that for any cyclic Steiner triple systems of order $v$ with $v\equiv 1\pmod{6}$, it is always possible to choose one block from each block orbit so that the chosen blocks are pairwise disjoint. We consider the generalization of this conjecture to cyclic $(v,k,λ)$-designs with $1 \leq λ\leq k-1$. Superimposing multiple copies of a cyclic symmetric design shows that the generalization cannot hold for all $v$, but we conjecture that it holds whenever $v$ is sufficiently large compared to $k$. We confirm that the generalization of the conjecture holds when $v$ is prime and $λ=1$ and also when $λ\leq (k-1)/2$ and $v$ is sufficiently large compared to $k$. As a corollary, we show that for any $k \geq 3$, with the possible exception of finitely many composite orders $v$, every cyclic $(v,k,1)$-design without short orbits is generated by a $(v,k,1)$-disjoint difference family.
