arXiv 2108.106572021-08-24math.CO

关于图的色边稳定指数

On the chromatic edge stability index of graphs

Saieed Akbari, Arash Beikmohammadi, Boštjan Brešar, Tanja Dravec, Mohammad Mahdi Habibollahi, Nazanin Movarraei

备注：: 17 pages

摘要：

给定一个非平凡图$G$，$G$中一个边集$F$的最小基数使得$χ'(G \setminus F)<χ'(G)$称为$G$的色边稳定性指数，记为$es_{χ'}(G)$，这样一个（最小）集$F$称为（最小）缓和集。当$1\le es_{χ'}(G)\le \lfloor n/2\rfloor$对任何图$G$成立时，我们研究了具有$es_{χ'}(G)$的极值和近极值的图.对具有$es_{χ'}(G)=\lfloor n/2\rfloor$的图$G$进行了分类，刻画了具有$es_{χ'}(G)=\lfloor n/2\rfloor-1$和$χ'(G)=Δ(G)+1$的图$G$.证明了奇圈和$K_2$是正则连通图，其边色稳定性指标为$1$ ;另一方面，证明了判定图$G$是否具有$es_{χ'}(G)=1$是NP-困难的.我们还证明了$r$ -正则图$G$的每一个最小缓和集，其中$r\ne 4$与$es_{χ'}(G)=2$是匹配的。此外，我们提出了一个猜想，每个图$G$存在一个最小缓解集，这是一个匹配，并证明了猜想成立的图$G$与$es_{χ'}(G)\in\{1,2,\lfloor n/2\rfloor-1,\lfloor n/2\rfloor\}$，并为二部图。

英文摘要：

Given a non-trivial graph $G$, the minimum cardinality of a set of edges $F$ in $G$ such that $χ'(G \setminus F)<χ'(G)$ is called the chromatic edge stability index of $G$, denoted by $es_{χ'}(G)$, and such a (smallest) set $F$ is called a (minimum) mitigating set. While $1\le es_{χ'}(G)\le \lfloor n/2\rfloor$ holds for any graph $G$, we investigate the graphs with extremal and near-extremal values of $es_{χ'}(G)$. The graphs $G$ with $es_{χ'}(G)=\lfloor n/2\rfloor$ are classified, and the graphs $G$ with $es_{χ'}(G)=\lfloor n/2\rfloor-1$ and $χ'(G)=Δ(G)+1$ are characterized. We establish that the odd cycles and $K_2$ are exactly the regular connected graphs with the chromatic edge stability index $1$; on the other hand, we prove that it is NP-hard to verify whether a graph $G$ has $es_{χ'}(G)=1$. We also prove that every minimum mitigating set of an $r$-regular graph $G$, where $r\ne 4$, with $es_{χ'}(G)=2$ is a matching. Furthermore, we propose a conjecture that for every graph $G$ there exists a minimum mitigating set, which is a matching, and prove that the conjecture holds for graphs $G$ with $es_{χ'}(G)\in\{1,2,\lfloor n/2\rfloor-1,\lfloor n/2\rfloor\}$, and for bipartite graphs.

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。

摘要：

英文摘要：

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