报告题目 (Title)：Flows of mono-directed signed graphs（单有向符号图上的流）
报告人 (Speaker)：李佳傲 副教授（南开大学）
报告时间 (Time)：2023年7月7日(周五) 19:00
报告地点 (Place)：B308，腾讯会议号693 612 919
邀请人(Inviter)：杨倩倩
报告摘要：A circular r-coloring of a graph G is a mapping \varphi: V(G)\mapsto [0,r) such that |\varphi(u)-\varphi(v)|\in[1,r-1] for each edge uv\in E(G). A natural Planar Circular Coloring Conjecture states that every planar graph of girth at least 2k is circular (2+2/k)-colorable. The k=1,2 cases are known as the 4CT and Grotzsch's theorem(3CT). It is open for k\ge 3, and the general girth condition is known for roughly 3k. Recently, the concept of circular coloring in signed graph was introduced by R. Naserasr, Z. Wang, and X. Zhu[E-JC 2021]. For a signed graph (G,\sigma), a circular r-coloring \varphi: V(G)\mapsto [0,r) applies the same rule for the positive edges, and it satisfies |\varphi(x)-\varphi(y)|\in[0,\frac{r}{2}-1]\cup [\frac{r}{2}+1, r) for each negative edge xy\in E(G). In this talk, we will discuss the circular coloring of signed planar graphs with given girth conditions. Specifically, we show that every signed planar graph of girth at least 3k+2 is circular (2+2/k)-colorable, and that every signed bipartite planar graph of girth at least 6t-2 admits a homomorphism to the negative even cycle C_{-2t}. In fact, those results follow from the dual of more general flow results of mono-directed signed graphs in a recent joint work with R. Naserasr, Z. Wang, and X. Zhu.