The proof of a conjecture on the lewin number of primitive non-powerful signed digraphs

Suppose S is a primitive non-powerful signed digraph. A pair of SSSD walks of S are two directed walks of S, which have the same initial vertex, same terminal vertex and same length, but different signs. The lewin number of S, denoted l(S), is the least positive integer k such that there are both SSSD walks of lengths k and k+1 from some vertex u to some vertex v (possibly u again) of S. This paper presents a proof of a conjecture on l(S), which was put forward by You et al. [L.H. You, M.H. Liu, B.L. Liu, Bounds on the lewin number for primitive non-powerful signed digraphs, Acta Math. Appl. Sinica 35 (2012) 396–407].