### Citations

2828 |
The Art of Computer Programming
- Knuth
- 1998
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Citation Context ...uence of integers from [n] = {1, 2, . . . , n} with the property that each ksubset of [n] appears exactly once consecutively in the sequence. For example, 1234524135 is a universal cycle for pairs of =-=[5]-=-. Chung, et al. [1], defined universal cycles for a general class of combinatorial structures, generalizing both deBruijn sequences and Gray codes (see [7]). The books [5, 6] contain a wealth of infor... |

123 | A survey of combinatorial Gray codes
- Savage
- 1997
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Citation Context ...1234524135 is a universal cycle for pairs of [5]. Chung, et al. [1], defined universal cycles for a general class of combinatorial structures, generalizing both deBruijn sequences and Gray codes (see =-=[7]-=-). The books [5, 6] contain a wealth of information about generating such things efficiently. A necessary condition for the existence of an (n, k)-UCS is that n | (n k ) . This is because symmetry dem... |

63 | personal communication - Jackson, Caudill - 1996 |

57 | Universal cycles for combinatorial structures
- Chung, Diaconis, et al.
- 1992
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Citation Context ...rom [n] = {1, 2, . . . , n} with the property that each ksubset of [n] appears exactly once consecutively in the sequence. For example, 1234524135 is a universal cycle for pairs of [5]. Chung, et al. =-=[1]-=-, defined universal cycles for a general class of combinatorial structures, generalizing both deBruijn sequences and Gray codes (see [7]). The books [5, 6] contain a wealth of information about genera... |

26 |
Universal cycles of k-subsets and k-permutations.
- Jackson
- 1993
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Citation Context ...xist if and only if n divides ( n k ) . Progress on the conjecture has been slow. The k = 2 case is trivial, corresponding to the existence of eulerian circuits in Kn if and only if n is odd. Jackson =-=[3]-=- proved the conjecture for k = 3, and constructed ucyles for k = 4 and odd n, leaving the case n ≡ 2 mod 8 unresolved. In [2] we find the following result. Result 2 [2] Let n0(3) = 8, n0(4) = 9, and n... |

22 | Combinatorial generation
- Ruskey
(Show Context)
Citation Context ...universal cycle for pairs of [5]. Chung, et al. [1], defined universal cycles for a general class of combinatorial structures, generalizing both deBruijn sequences and Gray codes (see [7]). The books =-=[5, 6]-=- contain a wealth of information about generating such things efficiently. A necessary condition for the existence of an (n, k)-UCS is that n | (n k ) . This is because symmetry demands that each symb... |

13 | 1994),“On Universal Cycles of k-subsets of an n-set
- Hurlbert
(Show Context)
Citation Context ...tee that the union of Tn,k(C1) and Tn,k(C2) will be connected. It could happen, for example, that each Ci has two components that connect, resulting in two components for C1∪C2. However, as proven in =-=[2]-=-, if Hn,k is connected, then the union over all classes produces a connected Tn,k. We will clarify this with the map κ, defined as follows. Let C = [cp11 , . . . , c pt−1 t−1 ; ct] be a class, where t... |

2 |
Solution of an outstanding conjecture: the nonexistence of universal cycles with k = n− 2, Discrete Math
- Stevens, Buskell, et al.
- 2002
(Show Context)
Citation Context ...n (n, k)-UCs exist for k = 3, 4, and 6 with n ≥ n0(k) and gcd(n, k) = 1. Note that n0(k) = 3k suffices for k ∈ {3, 4, 6}. It would be nice to lower 3k as much as possible. To this end, Stevens et al. =-=[8]-=- proved the following. Result 3 [8] No (k + 2, k)-UC exists for k ≥ 2. Combined with particular computer examples found by Jackson [4] (e.g. (n, k) = (10, 4)), this suggests that n0(k) = k + 3 may suf... |