PIMS- UVic Discrete Math Seminar: Zhouningxin Wang

Given positive integers $p,q$ where $p$ is even and $p\geq 2q$, a circular $\frac{p}{q}$-flow in a mono-directed signed graph $(G, \sigma)$ is a pair $(D, f)$ where $D$ is an orientation on $G$ and $f: E(G)\to \mathbb{Z}$ satisfies that for each positive edge $e$, $q\leq |f(e)|\leq p-q$ and for each negative edge $e$, either $0\leq |f(e)|\leq \frac{p}{2}-q$ or $\frac{p}{2}+q\leq |f(e)|\leq p-1$, and the total in-flow equals the total out-flow at each vertex. This is the dual notion of circular $\frac{p}{q}$-coloring of signed graphs recently introduced in ``Circular chromatic number of signed graphs. R. Naserasr, Z. Wang, and X. Zhu. Electronic Journal of Combinatorics, 28(2)(2021), \#P2.44''. In this talk, we consider bipartite analogs of Jaeger's circular flow conjecture and its dual, Jaeger-Zhang conjecture. We show that every $(6k-2)$-edge-connected Eulerian signed graph admits a circular $\frac{4k}{2k-1}$-flow and every signed bipartite planar graph of negative-girth at least $6k-2$ admits a circular $\frac{4k}{2k-1}$-coloring. We also provide some recent results about the circular flow index of signed graphs with high edge-connectivities. This is joint work with Jiaao Li, Reza Naserasr, and Xuding Zhu.