A signed graph G is a graph associated with a mapping σ: E(G) →{+1, −1}. Let G be a graph/signed graph with an orientation on each edge and let A be an abelian group. A function f, from the edge set E(G) of G to the nonzero elements of A, is call an A-flow of G if at each vertex v ∈ V (G), the sum of f(e) over every e with head v is equal to the sum of f(e) over every e with tail v. Tutte conjectured that if a graph has a Z-flow, then it has a Z-flow f such tat |f(e)| ≤ 4 for each e ∈ E(G), which is related to graphs embedded in orientable surfaces. Bouchet conjectured that if a signed graph has a Z-flow, then it has a Z-flow f such tat |f(e)| ≤ 5 for each e ∈ E(G), which is related to graphs embedded in nonorientable surfaces. The theory of flows has a strong connection with the coloring of graphs. In this talk, we focus on recent results on flows of signed graphs.

范更华，福州大学教授。1988年获加拿大滑铁卢大学博士学位。入选中国科学院1996年度“百人计划”从美国亚利桑那州立大学回国工作。范更华主要从事图论基础理论及其应用研究，致力于图论在芯片设计EDA软件中的应用。获1998年度国家杰出青年科学基金资助、2005年度国家自然科学二等奖（独立完成人）、2022年度教育部自然科学一等奖（第一完成人）；曾任福州大学副校长、中国数学会组合数学与图论专业委员会主任、全国组合数学与图论研究会理事长、中国运筹学会副理事长；现任福州大学离散数学及其应用教育部重点实验室主任、福建省数学会理事长。自1997年起，一直担任国际图论界权威期刊《Journal of Graph Theory》执行主编（Managing Editor）。