Base of the non-powerful signed tournament

Abstract

A signed digraph $S$ is the digraph $D$ by assigning signs $1$ or $-1$ to each arc of $D$. The base of $S$ is the minimum number $k$ such that there is a pair walks which have the same initial and terminal point with length $k$, but different signs. In this paper we show that for $n\ge 5$ the upper bound of the base of a primitive non-powerful signed tournament $S_n$, which is the signed digraph by assigning $1$ or $-1$ to each arc of a primitive tournament $T_n$, is $\mathrm{m}\mathrm{a}\mathrm{x}\{2n+2,n+11\}$. Moreover we show that it is extremal except when $n=5,7$.