A connected digraph in which the in-degree of any vertex equals its out-degree is Eulerian; this baseline result is used as the basis of existence proofs for universal cycles (also known as ucycles or generalized deBruijn cycles or U-cycles) of several combinatorial objects. The existence of ucycles is often dependent on the specific representation that we use for the combinatorial objects. For example, should we represent the subset $\{2,5\}$ of $\{1,2,3,4,5\}$ as “25” in a linear string? Is the representation “52” acceptable? Or is it tactically advantageous (and acceptable) to go with $\{0,1,0,0,1\}$? In this paper, we represent combinatorial objects as graphs, as in [3], and exhibit the flexibility and power of this representation to produce graph universal cycles, or Gucycles, for k-subsets of an n-set; permutations (and classes of permutations) of $[n]=\{1,2,\dots ,n\}$, and partitions of an n-set, thus revisiting the classes first studied in [5]. Under this graphical scheme, we will represent $\{2,5\}$ as the subgraph A of $C_5$ with edge set consisting of $\{2,3\}$ and $\{5,1\}$, namely the “second” and “fifth” edges in $C_5$. Permutations are represented via their permutation graphs, and set partitions through disjoint unions of complete graphs.

任何顶点的入度等于其出度的连通图是欧拉式的；该基线结果用作多个组合对象的通用循环（也称为u循环或广义deBruijn循环或U循环）存在性证明的基础。ucycle的存在通常取决于我们用于组合对象的特定表示形式。例如，我们应该代表子集吗$\{2，5\}$ 的 $\{\mathrm{1个}，2，3，4，5\}$如线性字符串中的“ 25”？表示“ 52”是否可以接受？还是在战术上有利（并且可以接受）$\{0，\mathrm{1个}，0，0，\mathrm{1个}\}$？在本文中，我们将组合对象表示为图，如[3]所示，并展示了这种表示的灵活性和强大功能，可为n个集合的k个子集生成图通用周期或Gucycles。的排列（和排列类别）$[ñ]=\{\mathrm{1个}，2，\dots ，ñ\}$以及n集的分区，从而重新研究了[5]中首次研究的类。在此图形方案下，我们将表示$\{2，5\}$作为子图阿的$C_5$ 边集包括 $\{2，3\}$ 和 $\{5，\mathrm{1个}\}$，即位于的“第二”和“第五”边缘 $C_5$。排列通过其排列图表示，并通过完整图的不相交并集来设置分区。