Nowhere-zero flows on signed regular graphs

We study the flow spectrum S(G) and the integer flow spectrum S(G) of odd regular graphs. 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 [7]. Let G be a (2t + 1)-regular graph. We show that if r ∈ S(G), then r = 2 + 1t or r ≥ 2 + 2 2t−1 . This result generalizes a result of [10] to signed graphs. In order to generalize Tutte’s characterization of bipartite cubic graphs [12], we introduce the notion of r-minimal sets for r ≥ 2. A set X ⊆ E(G) is a minimal set such that G −X is bipartite if and only if X is (2 + 1t )-minimal. Furthermore, 2 + 1 t is an element of the flow spectrum of a (2t+ 1)-regular graph 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 (2t + 1)-regular graph (H,σ) with 3 ∈ S(H) and H does not have a 1-factor. Let G (6= K3 2 ) be a cubic graph. If G 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). (3) 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 prove lower and upper bounds for the cardinality of smallest 3-minimal and 4-minimal sets of cubic graphs, respectively. Bouchet’s conjecture [2], that every flow-admissible signed graph admits a nowhere-zero 6-flow is equivalent to its restriction on cubic ∗Fellow of the International Graduate School ”Dynamic Intelligent Systems” †Paderborn Institute for Advanced Studies in Computer Science and Engineering, Paderborn University, Warburger Str. 100, 33102 Paderborn, Germany; [email protected]; [email protected] 1 ar X iv :1 30 7. 15 62 v3 [ m at h. C O ] 2 6 Se p 20 13 graphs. The paper concludes with a proof of Bouchet’s conjecture for