报告题目：Locally bi-2-transitive graphs and cycle-regular graphs
报告人：周进鑫教授（北京交通大学）
报告时间：2022年10月28日（周五） 14:00-15:00
腾讯会议：会议 ID：815-504-273
摘要：A vertex-transitive but not edge-transitive graph $G$ is called {\em locally bi-$2$-transitive\/} if the stabiliser $S$ in the full automorphism group of $G$ of every vertex $v$ of $G$ has two orbits of equal size on the neighbourhood of $v$, and $S$ acts $2$-transitively on each of these two orbits. Also a graph is called {\em cycle-regular\/} if the number of cycles of a given length passing through a given edge in the graph is a constant, and a graph with girth $g$ is called {\em edge-girth-regular\/} if the number of cycles of length $g$ passing through any edge in the graph is a constant.
In this talk, I shall discuss the characterization of edge-girth-regular and locally bi-$2$-transitive graphs of girth $3$. It is proved that a graph of girth $3$ is edge-girth-regular and locally bi-$2$-transitive if and only if $\G$ is the line graph of a semi-symmetric locally $3$-transitive graph. Then as an application, we prove that every tetravalent edge-girth-regular locally bi-$2$-transitive graph of girth $3$ is cycle-regular. This shows that vertex-transitive cycle-regular graphs need not to be edge-transitive, and hence resolves the problem posed by Fouquet and Hahn at the end of their paper `Cycle regular graphs need not be transitive', in {\em Discrete Appl. Math.} 113 (2001) 261--264.
报告人简介：周进鑫，北京交通大学教授，主要从事图的对称性研究。先后主持完成国家青年基金一项，面上项目两项，目前正在主持面上项目一项。在组合及图论领域权威JCTA/B,Combinatorica,JGT,JACO，EJC等期刊发表论文60余篇。