arXiv 1902.092602019-02math.CO

匹配覆盖图中的等价类

Equivalence classes in matching covered graphs

Fuliang Lu, Nishad Kothari, Xing Feng, Lianzhu Zhang

摘要：

一个二阶或二阶以上的连通图$G$是匹配覆盖的，如果每条边都位于某个\pema中。匹配覆盖图$G$的紧割分解产生一个砖和大括号的列表;根据Lov{á}sz~ \cite{lova87}的定理，这个列表是唯一的（至多多条边）; $b(G)$表示砖的数量，$c_4(G)$表示与循环$C_4$同构的大括号的数量（至多多条边）。两条边$e$和$f$是相互依赖的，如果对于每个完美匹配的$M$，$e \in M$当且仅当$f \in M$ ; Carvalho，Lucchesi和Murty在他们的里程碑式的论文~ \cite{clm99}中研究了这个概念。对于任何匹配覆盖图$G$，相互依赖是一个等价关系，它将$E(G)$划分为等价类;这个等价类划分用$\mathcal{E}_G$表示，我们将其部分称为$G$的等价类;我们使用$\varepsilon(G)$表示最大等价类的基数。“拼接”操作可以用来从较小的图构造较大的匹配覆盖图;参见~ \cite{lckm18} ;“紧密拼接”是“拼接”的一个更强版本。（这些是“分离切割”和“紧密切割”概念的对话。）在这篇文章中，我们回答了以下基本问题：如果匹配覆盖图$G$是通过“拼接”（或通过“紧密拼接”）两个较小的匹配覆盖图，比如~ $G_1$ ~和~ $G_2$得到的，那么$\mathcal{E}_G$与$\mathcal{E}_{G_1}$和$\mathcal{E}_{G_2}$是如何相关的（反之亦然）？作为我们研究结果的应用：首先，我们根据$b(G)$和$c_4(G)$建立了$\varepsilon(G)$的严格上界;其次，我们通过构造同时具有任意高$κ(G)$ ~和~ $\varepsilon(G)$的图，肯定地回答了He、Wei、Ye和Zhai最近提出的问题~ \cite{hwyz19}，其中$κ(G)$表示顶点连通性。

英文摘要：

A connected graph $G$, of order two or more, is matching covered if each edge lies in some \pema. The tight cut decomposition of a matching covered graph $G$ yields a list of bricks and braces; as per a theorem of Lov{á}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 perfect matching $M$, $e \in M$ if and only if $f \in M$; Carvalho, Lucchesi and Murty investigated this notion in their landmark paper~\cite{clm99}. For any matching covered graph $G$, mutual dependence is an equivalence relation, and it partitions $E(G)$ into equivalence classes; this equivalence class partition is denoted by $\mathcal{E}_G$ and we refer to its parts as equivalence classes of $G$; we use $\varepsilon(G)$ to denote the cardinality of the largest equivalence class. The operation of `splicing' may be used to construct bigger matching covered graphs 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 matching covered graph $G$ is obtained by `splicing' (or by `tight splicing') two smaller matching covered graphs, 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 vice versa)? 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 $κ(G)$~and~$\varepsilon(G)$ simultaneously, where $κ(G)$ denotes the vertex-connectivity.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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