From Squashed 6‐Cycles to Steiner Triple Systems

Lindner, Charles C.; Meszka, Mariusz; Rosa, Alexander

doi:10.1002/jcd.21346

Cited by 5 publications

(29 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(ii) For each group g ∈ G of size 2, define a copy of a max packing of K 7 with 6-cycles, with vertex set {∞ 1 , ∞ 2 , ∞ 3 }∪(b×{1, 2}), that can be squashed into 6-triples with leave (∞ 1 , ∞ 2 , ∞ 3 ) [11]. Add these 6-cycles to C. Then (X, C) is a max packing of K 12k+11 with 6-cycles with leave L in (i).…”

Section: N ≡ 11 (Mod 12)mentioning

confidence: 99%

“…(ii) For each group g of size 2, define a copy of Example (ii) For each group g of size 3, place a copy of a 6-cycle system of order 13 which can be squashed into a triple system [11] (no leave) on {∞} ∪ (g × {1, 2, 3, 4}) and place these 6-cycles in C. Then (X, C) is a max packing of K 12k+5 with 6-cycles with leave the 4-cycle in (i). Squashing the 6-cycles in (i), (ii) and (iii) produces a max packing of K 12k+5 with triples with leave the 4-cycle in (i).…”

Section: N ≡ 5 (Mod 12)mentioning

confidence: 99%

“…Therefore it makes sense to ask the following question: what is the spectrum for 6-cycle systems that can be squashed into Steiner triple systems? In [11], a complete solution is given to this problem by constructing for every n ≡ 1 or 9 (mod 12) a 6-cycle system that can be squashed into a Steiner triple system. Example 1.1.…”

Section: Introductionmentioning

confidence: 99%

“…Example 1.1. (A 6-cycle system of order 9 squashed into a triple system [11].) Now if n ≡ 3 or 7 (mod 12) there does not exist a 6-cycle system of order n. However, there does exist a maximum packing (max packing) of K n with 6-cycles with leave a triple (i.e., a pair (X, C) and a set L, the leave, where C is a collection of edge disjoint 6-cycles with verteces in X, L is the set of the edges of K n not belonging to any 6-cycle of C and |L| is as small as possible) and so the following question makes sense.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Squashing maximum packings of 6-cycles into maximum packings of triples

2015

View full text Add to dashboard Cite

A 6-cycle is said to be squashed if we identify a pair of opposite vertices and name one of them with the other (and thereby turning the 6-cycle into a pair of triples with a common vertex). The squashing problem for 6-cycle systems was introduced by C. C. Lindner, M. Meszka and A. Rosa and completely solved by determining the spectrum. In this paper, by employing PBD and GDD-constructions and filling techniques, we extend this result by squashing maximum packings of K n with 6-cycles into maximum packings of K n with triples. More specifically, we establish that for each n ≥ 6, there is a max packing of K n with 6-cycles that can be squashed into a maximum packing of K n with triples.

show abstract

Section: N ≡ 11 (Mod 12)mentioning

confidence: 99%

Section: N ≡ 5 (Mod 12)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Squashing maximum packings of 6-cycles into maximum packings of triples

2015

View full text Add to dashboard Cite

show abstract

“…Quite recently a new connection between 6-cycle systems and Steiner triple systems was introduced: the squashing of a 6-cycle system into a Steiner triple system [9]. A definition is in order.…”

Section: Introductionmentioning

confidence: 99%

Squashing maximum packings of Kn with 8-cycles into maximum packings of Kn with 4-cycles

Lindner

Faro

Meszka

et al. 2014

Filomat

Self Cite

View full text Add to dashboard Cite

An 8-cycle is said to be squashed if we identify a pair of opposite vertices and name one of them with the other (and thereby turning the 8-cycle into a pair of 4-cycles with exactly one vertex in common). The resulting pair of 4-cycles is called a bowtie. We say that we have squashed the 8-cycle into a bowtie. Evidently an 8-cycle can be squashed into a bowtie in eight different ways. The object of this paper is the construction, for every n > 8, of a maximum packing of Kn with 8-cycles which can be squashed in a maximum packing of Kn with 4-cycles.

show abstract

Squashing minimum coverings of 6-cycles into minimum coverings of triples

et al. 2014

Self Cite

View full text Add to dashboard Cite

A 6-cycle is said to be squashed if we identify a pair of opposite vertices and name one of them with the other (thereby turning the 6-cycle into a pair of triples with a common vertex). A 6-cycle can be squashed in six different ways. The spectrum for 6-cycle systems that can be squashed into Steiner triple systems has been determined by Lindner, Meszka and Rosa [From squashed 6-cycles to Steiner triple systems, J. Combin. Des. [6]]. The squashing of maximum packings of K n with 6-cycles into maximum packings of K n with triples has also been fully dealt with by Lindner, Lo Faro and Tripodi [Squashing maximum packings of 6-cycles into maximum packings of triples, submitted]. The object of this paper is to extend these results to minimum coverings of Kn with 6-cycles into minimum coverings of Kn with triples. We give a complete solution.

show abstract

From Squashed 6‐Cycles to Steiner Triple Systems

Abstract: Abstract:Squashed 6-cycle systems are introduced as a natural counterpart to 2-perfect 6-cycle systems. The spectrum for the latter has been determined previously in [5]. We determine completely the spectrum for squashed 6-cycle systems, and also for squashed 6-cycle packings.

Cited by 5 publications

References 3 publications

Squashing maximum packings of 6-cycles into maximum packings of triples

Squashing maximum packings of 6-cycles into maximum packings of triples

Squashing maximum packings of Kn with 8-cycles into maximum packings of Kn with 4-cycles

Squashing minimum coverings of 6-cycles into minimum coverings of triples

Contact Info

Product

Resources

About