The Number of Nowhere-Zero Flows on Graphs and Signed Graphs

A nowhere-zero $k$-flow on a graph $\Gamma$ is a mapping from the edges of $\Gamma$ to the set \{\pm1, \pm2, ..., \pm(k-1)\} \subset \bbZ such that, in any fixed orientation of $\Gamma$, at each node the sum of the labels over the edges pointing towards the node equals the sum over the edges pointing away from the node. We show that the existence of an \emph{integral flow polynomial} that counts nowhere-zero $k$-flows on a graph, due to Kochol, is a consequence of a general theory of inside-out polytopes. The same holds for flows on signed graphs. We develop these theories, as well as the related counting theory of nowhere-zero flows on a signed graph with values in an abelian group of odd order. Our results are of two kinds: polynomiality or quasipolynomiality of the flow counting functions, and reciprocity laws that interpret the evaluations of the flow polynomials at negative integers in terms of the combinatorics of the graph.Comment: 17 pages, to appear in J. Combinatorial Th. Ser.

Signed circuit 66-covers of signed K4K_4-minor-free graphs

Bermond, Jackson and Jaeger [{\em J. Combin. Theory Ser. B} 35 (1983): 297-308] proved that every bridgeless ordinary graph $G$ has a circuit $4$-cover and Fan [{\em J. Combin. Theory Ser. B} 54 (1992): 113-122] showed that $G$ has a circuit $6$-cover which together implies that $G$ has a circuit $k$-cover for every even integer $k\ge 4$. The only left case when $k = 2$ is the well-know circuit double cover conjecture. For signed circuit $k$-cover of signed graphs, it is known that for every integer $k\leq 5$, there are infinitely many coverable signed graphs without signed circuit $k$-cover and there are signed eulerian graphs that admit nowhere-zero $2$-flow but don't admit a signed circuit $1$-cover. Fan conjectured that every coverable signed graph has a signed circuit $6$-cover. This conjecture was verified only for signed eulerian graphs and for signed graphs whose bridgeless-blocks are eulerian. In this paper, we prove that this conjecture holds for signed $K_4$-minor-free graphs. The $6$-cover is best possible for signed $K_4$-minor-free graphs