Characteristic flows on signed graphs and short circuit covers (Q311542)

Summary: We generalise to signed graphs a classical result of \textit{W. T. Tutte} [Can. J. Math. 8, 13--28 (1956; Zbl 0070.02302)] stating that every integer flow can be expressed as a sum of characteristic flows of circuits. In our generalisation, the rôle of circuits is taken over by signed circuits of a signed graph which occur in two types -- either balanced circuits or pairs of disjoint unbalanced circuits connected with a path intersecting them only at its ends. As an application of this result we show that a signed graph \(G\) admitting a nowhere-zero \(k\)-flow has a covering with signed circuits of total length at most \(2(k-1)|E(G)|\).