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Graph universal cycles of combinatorial objects
Advances in Applied Mathematics ( IF 1.0 ) Pub Date : 2021-01-26 , DOI: 10.1016/j.aam.2021.102166
Amelia Cantwell , Juliann Geraci , Anant Godbole , Cristobal Padilla

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,,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 C5 with edge set consisting of {2,3} and {5,1}, namely the “second” and “fifth” edges in C5. Permutations are represented via their permutation graphs, and set partitions through disjoint unions of complete graphs.



中文翻译:

图组合对象的通用循环

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{25}{1个2345}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){01个001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Gucycles。的排列(和排列类别)[ñ]={1个2ñ}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{25}作为子图C5 边集包括 {23}{51个},即位于的“第二”和“第五”边缘 C5。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。

更新日期:2021-01-28
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