# 12月18日 Songling Shan博士學術報告（數學與統計學院）

Songling Shan 博士于2014年畢業于Georgia State University, 獲理學博士學位，師從陳冠濤教授，主要研究方向是圖論及組合，具體包括圖的結構，圖染色及極值圖論。2015年至2018年在Vanderbilt University 做 Mark Ellingham 教授的博士后。現在是Illinois State University 助理教授。已在J. Graph Theory, SIAM J. Discrete Math., J.Combin. Theory Ser. B 等國際期刊發表學術論文20余篇。

Let $G$ be a simple graph with maximum degree $\Delta$.  We call $G$ \emph{overfull} if $|E(G)|>\Delta \lfloor |V(G)|/2\rfloor$. The \emph{core} of $G$, denoted $G_{\Delta}$, is the subgraph of $G$ induced by its vertices of degree $\Delta$. A classical result of Vizing shows that $\chi'(G)$, the chromatic index of $G$, is either $\Delta$ or $\Delta+1$.   It is NP-complete to determine  the chromatic index  for a general graph.   However, $\chi'(G)=\Delta+1$ if $G$ is overfull. Hilton and Zhao in 1996 conjectured  that if $G$ is a simple connected graph with $\Delta\ge 3$ and  $\Delta(G_\Delta)\le 2$, then $\chi'(G)=\Delta+1$ if and only if $G$ is overfull or $G=P^*$, where $P^*$ is obtained from the Petersen graph by deleting a vertex. The progress on the conjecture has been slow: it was only confirmed for $\Delta=3,4$, respectively, in 2003 and 2017. In this talk, we will take about the confirmation of this conjecture for all $\Delta\ge 4$ and the more general overfull graph conjecture.

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