THE BASES OF PRIMITIVE NON-POWERFUL COMPLETE SIGNED GRAPHS

The base of a signed digraph S is the minimum number k such that for any vertices u, v of S, there is a pair of walks of length k from u to v with different signs. Let K be a signed complete graph of order n, which is a signed digraph obtained by assigning +1 or -1 to each arc of the n-th order complete graph $K_n$ considered as a digraph. In this paper we show that for $n{\geq}3$ the base of a primitive non-powerful signed complete graph K of order n is 2, 3 or 4.