Nowhere-zero flows on signed regular graphs

We study the flow spectrum S ( G ) and the integer flow spectrum S ? ( G ) of signed ( 2 t + 1 ) -regular graphs. We show that if r ? S ( G ) , then r = 2 + 1 t or r ? 2 + 2 2 t - 1 . Furthermore, 2 + 1 t ? S ( G ) if and only if G has a t -factor. If G has a 1-factor, then 3 ? S ? ( G ) , and for every t ? 2 , there is a signed ( 2 t + 1 ) -regular graph ( H , ? ) with 3 ? S ? ( H ) and H does not have a 1-factor.If G ( ? K 2 3 ) is a cubic graph which has a 1-factor, then { 3 , 4 } ? S ( G ) ? S ? ( G ) . Furthermore, the following four statements are equivalent: (1) G has a 1-factor. (2) 3 ? S ( G ) . (3) 3 ? S ? ( G ) . (4) 4 ? S ? ( G ) . There are cubic graphs whose integer flow spectrum does not contain 5 or 6, and we construct an infinite family of bridgeless cubic graphs with integer flow spectrum { 3 , 4 , 6 } .We show that there are signed graphs where the difference between the integer flow number and the flow number is greater than or equal to 1, disproving a conjecture of Raspaud and Zhu.The paper concludes with a proof of Bouchet's 6-flow conjecture for Kotzig-graphs.