题目：Equivalence classes in matching covered graphs

报告摘要：

A connected graph~$G$, of order two or more, is \mc\ if each edge lies in some \pema.
The tight cut decomposition of a \mcg~$G$ yields a list of bricks and braces; as per a theorem
of Lov{\'a}sz~\cite{lova87}, this list is unique (up to multiple edges); $b(G)$ denotes
the number of bricks, and $c_4(G)$ denotes the number of braces that are isomorphic to
the cycle~$C_4$(up to multiple edges).
Two edges $e$ and $f$ are mutually dependent if, for each \pema~$M$,
$e \in M$ if and only if $f \in M$; Carvalho, Lucchesi and Murty investigated this notion in their landmark paper~\cite{clm99}.For any \mcg~$G$, mutual dependence is an equivalence relation, and it partitions $E(G)$ into {\ec}es; this \ecp\ is denoted by $\mathcal{E}_G$ and we refer to its parts as {\ec}es of~$G$; we use $\varepsilon(G)$ to denote the cardinality of the largest \ec.
The operation of splicing' may be used to construct bigger {\mcg}s from smaller ones;
see~\cite{lckm18}; tight splicing' is a stronger version of splicing'.
(These are converses of the notions of separating cut' and tight cut'.)
In this article, we answer the following basic question: if a \mcg~$G$ is obtained by splicing' (or by tight splicing') two smaller {\mcg}s, say~$G_1$~and~$G_2$, Then how is $\mathcal{E}_G$

related to~$\mathcal{E}_{G_1}$ and to~$\mathcal{E}_{G_2}$ (and viceversa)?
 
As applications of our findings: firstly, we establish tight upper bounds on $\varepsilon(G)$ in terms of $b(G)$

and $c_4(G)$; secondly, we answer a recent question of He, Wei,Ye and Zhai~\cite{hwyz19}, in the affirmative,

by constructing graphs that have arbitrarily high $\kappa(G)$~and~$\varepsilon(G)$ simultaneously, where $\kappa(G)$ denotes the vertex-connectivity.

主讲人：卢福良

报告时间：2021.5.20，10:30-11:30

报告地点：综合楼810会议室

个人简介：卢福良，博士，闽南师范大学教授，硕士生导师，福建省闽江特聘教授。
主要从事图的匹配理论研究，在SIAM J.Discrete Math.、Journal of Graph Theory、Electron.J.Comb.、Discrete Mathematics等国内外主要期刊发表论文30余篇。