Nowhere-zero flows on signed regular graphs

• DOI: 10.1016/j.ejc.2015.02.007
• Publication Date: 2015-03-07T01:01:34Z
• Authors: Michael Schubert; Eckhard Steffen
• Keywords: FOS: Computer and information sciences; Discrete Mathematics (cs.DM); 05C21, 05C22; FOS: Mathematics; Mathematics - Combinatorics; Combinatorics (math.CO); 0102 computer and information sciences; 01 natural sciences; Computer Science - Discrete Mathematics

We study the flow spectrum ${\cal S}(G)$ and the integer flow spectrum $\overline{\cal S}(G)$ of signed $(2t+1)$-regular graphs. We show that if $r \in {\cal S}(G)$, then $r = 2+\frac{1}{t}$ or $r \geq 2 + \frac{2}{2t-1}$. Furthermore, $2 + \frac{1}{t} \in {\cal S}(G)$ if and only if $G$ has a $t$-factor. If $G$ has a 1-factor, then $3 \in \overline{\cal S}(G)$, and for every $t \geq 2$, there is a signed $(2t+1)$-regular graph $(H,��)$ with $ 3 \in \overline{\cal S}(H)$ and $H$ does not have a 1-factor. If $G$ $(\not = K_2^3)$ is a cubic graph which has a 1-factor, then $\{3,4\} \subseteq {\cal S}(G) \cap \overline{\cal S}(G)$. Furthermore, the following four statements are equivalent: (1) $G$ has a 1-factor. (2) $3 \in {\cal S}(G)$. (3) $3 \in \overline{\cal S}(G)$. (4) $4 \in \overline{\cal S}(G)$. 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.<br/>24 pages, 4 figures; final version; to appear in European J. Combinatorics<br/>