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 it it tactically advantageous (and acceptable) to go with $\{ 0,1,0,0,1\}$? In this paper, we represent combinatorial objects as graphs, as in \cite{ bks}, and exhibit the flexibility and power of this representation to produce { \it graph universal cycles}, or { \it Gucycles}, for $k$-subsets of an $n$-set; permutations (and classes of permutations) of $[n]=\{ 1,2,\ldots,n\}$, and partitions of an $n$-set, thus revisiting the classes first studied in \cite{ cdg}. 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.
Categories: math.CO