A uniformly-ordered $\rho$-labeling (also known as a ${\rho }^{++}$-labeling) of a bipartite graph was introduced by El-Zanati, Vanden Eynden, and Punnim. Such a labeling of a bipartite graph $G$ with $m$ edges yields a cyclic $G$-decomposition of $K_{2mt+1}$ for every positive integer $t$. Here we show that for every even integer $n\ge 8$, the generalized Petersen graph $P(n,3)$ admits a ${\rho }^{++}$-labeling and hence cyclically decomposes $K_{6nt+1}$ for all positive integers $t$.

中文翻译：

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关于完全图的循环分解成二部广义Petersen图 $P（ñ，3）$

均匀排序 $\rho$-标签（也称为 ${\rho }^{++}$El-Zanati，Vanden Eynden和Punnim引入了二部图的-标记）。二部图的这种标记$G$ 和 $米$ 边产生循环 $G$-分解 ${ķ}_{2米Ť+\mathrm{1个}}$ 对于每个正整数 $Ť$。在这里，我们表明对于每个偶数整数$ñ\ge 8$，广义Petersen图 $P（ñ，3）$ 承认 ${\rho }^{++}$标记，因此周期性分解 ${ķ}_{6ñŤ+\mathrm{1个}}$ 对于所有正整数 $Ť$。