Zero-sum flows in designs

✍ Scribed by S. Akbari; G. B. Khosrovshahi; A. Mofidi

• Publisher: John Wiley and Sons
• Year: 2011
• Tongue: English
• Weight: 114 KB
• Volume: 19
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.20289

✦ Synopsis

Let D be a t- (v, k,k) design and let N i (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 N 1 (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 nonsymmetric 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), (v,k,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. q 2011