论文标题
给定程度和周长的小图和超图
Small graphs and hypergraphs of given degree and girth
论文作者
论文摘要
搜索最小的$ d $ grath $ g $图表的历史悠久，通常被称为笼子问题。这个问题自然而然地扩展到了超图，我们可能会要求在$ d $的$ d $，$ r $ r $ - 均匀的超图（berge）girth $ g $中提供最小数量的顶点。我们表明，这两个问题实际上是密切相关的。通过将Cayley图的想法扩展到HyperGraph上下文，我们找到了各种参数集的最小已知超图。由于图理论中与笼子问题的紧密联系，我们能够使用这些技术来找到新的记录最小的围栏23、24、28、29、29、30、31和32的最小立方图。
The search for the smallest possible $d$-regular graph of girth $g$ has a long history, and is usually known as the cage problem. This problem has a natural extension to hypergraphs, where we may ask for the smallest number of vertices in a $d$-regular, $r$-uniform hypergraph of given (Berge) girth $g$. We show that these two problems are in fact very closely linked. By extending the ideas of Cayley graphs to the hypergraph context, we find smallest known hypergraphs for various parameter sets. Because of the close link to the cage problem from graph theory, we are able to use these techniques to find new record smallest cubic graphs of girths 23, 24, 28, 29, 30, 31 and 32.
