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arXiv 2103.08831math.CO
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正则饱和图与无和集

Regular saturated graphs and sum-free sets

Craig Timmons

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摘要：

在最近的一篇论文中，Gerbner，Patkós，Tuza和Vizer研究了正则$F$ -饱和图。其中一个基本问题是给出了$F$，对于$n$，是否存在正则的$n$ -顶点$F$ -饱和图.他们证明了对所有充分大的$n$，存在一个正则的$K_3$ -饱和图，其顶点为$n$.我们将这个结果推广到$K_4$和$K_5$，并证明了一些更大的完全图的部分结果。利用加法组合学中的无和集的一种变形，证明了对于所有的$k \geq 2$，存在一个正则的$C_{2k+1}$ --对于无穷多个$n$，其顶点被$n$饱和.研究和自由集，产生$C_{2k+1}$ -饱和图本身是一个有趣的问题，我们在这个方向上的一个开放的问题。

英文摘要：

In a recent paper, Gerbner, Patkós, Tuza and Vizer studied regular $F$-saturated graphs. One of the essential questions is given $F$, for which $n$ does a regular $n$-vertex $F$-saturated graph exist. They proved that for all sufficiently large $n$, there is a regular $K_3$-saturated graph with $n$ vertices. We extend this result to both $K_4$ and $K_5$ and prove some partial results for larger complete graphs. Using a variation of sum-free sets from additive combinatorics, we prove that for all $k \geq 2$, there is a regular $C_{2k+1}$-saturated with $n$ vertices for infinitely many $n$. Studying the sum-free sets that give rise to $C_{2k+1}$-saturated graphs is an interesting problem on its own and we state an open problem in this direction.

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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