Twofold triple systems without 2-intersecting Gray codes

Erzurumluoğlu, Aras; Pike, David A.

doi:10.1007/s10623-016-0259-7

Cited by 7 publications

(12 citation statements)

References 13 publications

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“…Mahmoodian observed that the unique (up to isomorphism) TTS on six points has the Petersen graph as its 2‐BIG , and even earlier Colbourn and Johnstone had demonstrated a TTS(19) with a connected non‐Hamiltonian 2‐BIG . It has recently been established that TTSs with connected non‐Hamiltonian 2‐BIGs exist for most admissible orders; in particular, the current authors have completely determined the spectrum for TTSs that do not have cyclic 2‐intersecting Gray codes but which nevertheless have connected 2‐BIGs . Theorem There exists a TTS( v ) with a connected non‐Hamiltonian 2‐BIG if and only if

v = 6

v ⩾ 12

v \equiv 0

or 1 (modulo 3).…”

Section: Introductionmentioning

confidence: 92%

“…Then a Hamilton cycle in the 2-BIG of  ′ is given as ( 1 , {(0, 1), (1, 1), (1, 3)}, 6 , {(1, 1), (0, 2), ( − 1, 2)}, 8 , {(0, 1), (1, 2), ( − 1, 2)}, 5 , {( − 1, 1), (0, 2), (1, 2)}, 2 , {( − 1, 2), (0, 3), ( − 1, 3)}, 9 , {(1, 2), (0, 3), (1, 3)}, 4 , {(1, 1), ( − 1, 1), (0, 3)}, 7 , {(0, 1), ( − 1, 1), ( − 1, 3)}, 3 , {(0, 2), (1, 3), ( − 1, 3)}). □…”

Section: + Construction When ≡ (Modulo 6)mentioning

confidence: 99%

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Twofold triple systems with cyclic 2‐intersecting Gray codes

Erzurumluoğlu

Pike²

2017

J of Combinatorial Designs

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Given a combinatorial design D with block set B, the block-intersection graph (BIG) of D is the graph that has B as its vertex set, where two vertices B1 ∈ B and B2 ∈ B are adjacent if and only if |B1 ∩ B2| > 0. The i-block-intersection graph (i-BIG) of D is the graph that has B as its vertex set, where two vertices B1 ∈ B and B2 ∈ B are adjacent if and only if |B1 ∩B2| = i. In this paper several constructions are obtained that start with twofold triple systems (TTSs) with Hamiltonian 2-BIGs and result in larger TTSs that also have Hamiltonian 2-BIGs. These constructions collectively enable us to determine the complete spectrum of TTSs with Hamiltonian 2-BIGs (equivalently TTSs with cyclic 2-intersecting Gray codes) as well as the complete spectrum for TTSs with 2-BIGs that have Hamilton paths (i.e., for TTSs with 2-intersecting Gray codes).In order to prove these spectrum results, we sometimes require ingredient TTSs that have large partial parallel classes; we prove lower bounds on the sizes of partial parallel clasess in arbitrary TTSs, and then construct larger TTSs with both cyclic 2-intersecting Gray codes and parallel classes.

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v = 6

v ⩾ 12

v \equiv 0

or 1 (modulo 3).…”

Section: Introductionmentioning

confidence: 92%

Section: + Construction When ≡ (Modulo 6)mentioning

confidence: 99%

Twofold triple systems with cyclic 2‐intersecting Gray codes

Erzurumluoğlu

Pike²

2017

J of Combinatorial Designs

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“…As already mentioned, the existence of TTSs with connected non-Hamiltonian 2-BIGs has previously been established [15]. A few of the constructions in [15] are able to produce bipartite 2-BIGs when a (smaller) bipartite 2-BIG is used as an ingredient. However, no such 2-BIGs were known to exist prior to our current paper.…”

Section: Introductionmentioning

confidence: 95%

“…Block intersection graphs of designs have been a focus of attention since Ron Graham pondered whether STS block intersection graphs might be Hamiltonian, which was subsequently confirmed [4,24]. For some additional results concerning Hamilton cycles and similar properties in block intersection graphs of designs, see [1,3,5,9,12,15,16,22,23,26,28,30,31,35,36].…”

Section: Introductionmentioning

confidence: 95%

“…It is known that for v 4 such that v ≡ 0, 1 (mod 3) and v = 6, there exists a TTS(v) whose 2-BIG is Hamiltonian [12,16]. There also exists a TTS(v) whose 2-BIG is connected but non-Hamiltonian when v = 6 or v 12 and v ≡ 0, 1 (mod 3) [15]. While the existence of TTSs in each case has been established, it is not yet known what properties of a TTS are sufficient for the existence of a Hamilton cycle in the 2-BIG.…”

Section: Introductionmentioning

confidence: 99%

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Decomposable Twofold Triple Systems with Non-Hamiltonian 2-Block Intersection Graphs

Cameron

Pike

2020

Electron. J. Combin.

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The $2$-block intersection graph ($2$-BIG) of a twofold triple system (TTS) is the graph whose vertex set is composed of the blocks of the TTS and two vertices are joined by an edge if the corresponding blocks intersect in exactly two elements. The $2$-BIGs are themselves interesting graphs: each component is cubic and $3$-connected, and a $2$-BIG is bipartite exactly when the TTS is decomposable to two Steiner triple systems. Any connected bipartite $2$-BIG with no Hamilton cycle is a further counter-example to a disproved conjecture posed by Tutte in 1971. Our main result is that there exists an integer $N$ such that for all $v\geq N$, if $v\equiv 1$ or $3\mod{6}$ then there exists a TTS($v$) whose $2$-BIG is bipartite and connected but not Hamiltonian. Furthermore, $13<N\leq 663$. Our approach is to construct a TTS($u$) whose $2$-BIG is connected bipartite and non-Hamiltonian and embed it within a TTS($v$) where $v>2u$ in such a way that, after a single trade, the $2$-BIG of the resulting TTS($v$) is bipartite connected and non-Hamiltonian.

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$${\mathrm{TS}}(v,\lambda )$$ with Cyclic 2-Intersecting Gray Codes: $$v\equiv 0$$ or $$4\pmod {12}$$

Asplund

Keranen

2020

Graphs and Combinatorics

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Twofold triple systems without 2-intersecting Gray codes

Cited by 7 publications

References 13 publications

Twofold triple systems with cyclic 2‐intersecting Gray codes

Twofold triple systems with cyclic 2‐intersecting Gray codes

Decomposable Twofold Triple Systems with Non-Hamiltonian 2-Block Intersection Graphs

$${\mathrm{TS}}(v,\lambda )$$ with Cyclic 2-Intersecting Gray Codes: $$v\equiv 0$$ or $$4\pmod {12}$$

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