一、报告题目：Some results on circular flows and group connectivity of graphs

报告人：李学良（南开大学）

时间：9月16号下午2：30--3：30

地点：腾讯会议：593-830-550, 链接：https://meeting.tencent.com/dm/5ep26FdrT1Mz

摘要：Tutte proposed integer flows when he studied the well-known 4-Color Problem in 1949.For further study of flow properties, Jaeger et al. (1992) introduced the concept of group connectivity as a nonhomogeneous analogue of integer flows, and Goddyn et al. (1998) proposed the definitions of circular flows and flow index which extend the range of flows to rational numbers. In this talk, we will discuss the existence of circular flow for regular Class I graphs and the equivalence of group connectivity on non-isomorphic groups with a same order.

二、报告题目：Integer Flows and Modulo Flows of Signed Graphs

报告人：李佳傲（南开大学）

时间：9月16号下午3：30--4：30

地点：腾讯会议：593-830-550, 链接：https://meeting.tencent.com/dm/5ep26FdrT1Mz

摘要：It was conjectured by Bouchet in 1983 that every flow-admissible signed graph admits a nowhere-zero integer 6-flow. Recently, we prove that every flow-admissible signed graph admits a balanced Z_2 × Z_3-flow, and apply it to obtain a nowhere-zero integer 11-flow for every such signed graph. For a better understanding of Bouchet’s conjecture, it is crucial to study on how to convert modulo flows to better integer flows. In this talk, we will show how to convert Z_2-, Z_3-flow to integer flows in the 11-flow theorem proof, and will also discuss some new results on converting Z_5-flow and unitary Z_{2k+1}-flows to certain integer flows. We show that every bridgeless signed graph with a nowhere-zero Z_5-flow admits a nowhere-zero integer 7-flow. However, it seems much more difficult to convert Z_{2k}-flows to (desired) integer flows in signed graphs. In particular, is it possible to covert Z_6-flows to integer k-flows for some small value k (which would be able to improve the current integer 11-flow theorem)?