论文标题
在Oberwolfach的单翼问题上，通过优雅的标签
On the Oberwolfach problem for single-flip $2$-factors via graceful labelings
论文作者
论文摘要
令$ f $为$ 2 $的订单$ v $图表。 Oberwolfach问题$ OP（F）$，于1967年开放，仍要求将$ k_v $分解成$ f $的副本。在本文中，我们表明$ op（f）$只要$ f $具有足够大的周期，可以符合给定的下限，此外，它具有单翼自动形态，这是一种既定的自动形态，可以反映出$ f $的一个周期。此外，我们证明了最小覆盖版本和问题的最大包装版本的结果。当$ k_v $的边缘具有多重性2时，我们也会显示出类似的结果，但是在这种情况下，我们不需要$ f $是单翼。 我们的方法使我们能够以良好的自动形态为Oberwolfach问题明确构建解决方案，与基于概率方法的一些最近的渐近结果相比，这些方法是非构建方法，这些方法是非构建的，并且不能以$ f $的订单提供$ f $的下层限制，从而保证了$ op（f）$ $（f）$。 我们的构造基于双加倍结构，该结构适用于$ 2 $定型图的优雅标签，并删除了顶点。我们表明，只要路径组件的长度足够大，这类图就优美。对于存在此类图的$α$标记的路径长度的下限要更好。
Let $F$ be a $2$-regular graph of order $v$. The Oberwolfach problem $OP(F)$, posed in 1967 and still open, asks for a decomposition of $K_v$ into copies of $F$. In this paper we show that $OP(F)$ has a solution whenever $F$ has a sufficiently large cycle which meets a given lower bound and, in addition, has a single-flip automorphism, which is an involutory automorphism acting as a reflection on exactly one of the cycles of $F$. Furthermore, we prove analogous results for the minimum covering version and the maximum packing version of the problem. We also show a similar result when the edges of $K_v$ have multiplicity 2, but in this case we do not require that $F$ be single-flip. Our approach allows us to explicitly construct solutions to the Oberwolfach Problem with well-behaved automorphisms, in contrast with some recent asymptotic results, based on probabilistic methods, which are nonconstructive and do not provide a lower bound on the order of $F$ that guarantees the solvability of $OP(F)$. Our constructions are based on a doubling construction which applies to graceful labelings of $2$-regular graphs with a vertex removed. We show that this class of graphs is graceful as long as the length of the path-component is sufficiently large. A much better lower bound on the length of the path is given for an $α$-labeling of such graphs to exist.
