Non-extendible latin parallelepipeds

Kochol, Martin

doi:10.1016/j.ipl.2012.08.014

Cited by 2 publications

(1 citation statement)

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“…For related embedding results in higher dimensions and connections with maximum distance separable codes, see Krotov and Sotnikova [33], and Potapov [37,39]. Various results on latin boxes that cannot be completed to latin cubes can be found in [7,8,18,24,30,29,31,32,34]. For further results on latin cubes, see [5,15,25,38] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Ryser Type Conditions for Extending Colorings of Triples

Bahmanian¹

2022

Preprint

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In 1951, Ryser showed that an n ˆn array L whose top left r ˆs subarray is filled with n different symbols, each occurring at most once in each row and at most once in each column, can be completed to a latin square of order n if and only if the number of occurrences of each symbol in L is at least r `s ´n. We prove a Ryser type result on extending partial coloring of 3-uniform hypergraphs. Let X, Y be finite sets with X Ĺ Y and |Y | " 0 pmod 3q. When can we extend a (proper) coloring of λ `X 3 ˘(all triples on a ground set X, each one being repeated λ times) to a coloring of λ `Y 3 ˘using the fewest possible number of colors? It is necessary that the number of triples of each color in `X 3 ˘is at least |X| ´2|Y |{3. Using hypergraph detachments (Combin. Probab. Comput. 21 (2012), 483-495), we establish a necessary and sufficient condition in terms of list coloring complete multigraphs. Using Häggkvist-Janssen's bound (Combin. Probab. Comput. 6 (1997), 295-313), we show that the number of triples of each color being at least |X|{2 ´|Y |{6 is sufficient. Finally we prove an Evans type result by showing that if |Y | ě 3|X|, then any q-coloring of any subset of λ `X 3 ˘can be embedded into a λ `|Y |´1 2 ˘-coloring of λ `Y 3 ˘as long as q ď λ `|Y |´1 2 ˘´λ `|X| 3 ˘{ t|X|{3u.

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Section: Introductionmentioning

confidence: 99%

Ryser Type Conditions for Extending Colorings of Triples

Bahmanian¹

2022

Preprint

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Toward a three-dimensional counterpart of Cruse’s theorem

Bahmanian

2024

Proc. Amer. Math. Soc.

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Completing partial latin squares is NP-complete. Motivated by Ryser’s theorem for latin rectangles, in 1974, Cruse found conditions that ensure a partial symmetric latin square of order m m can be embedded in a symmetric latin square of order n n . Loosely speaking, this results asserts that an n n -coloring of the edges of the complete m m -vertex graph K m K_m can be embedded in a one-factorization of K n K_n if and only if n n is even and the number of edges of each color is at least m − n / 2 m-n/2 . We establish necessary and sufficient conditions under which an edge-coloring of the complete λ \lambda -fold m m -vertex 3-graph λ K m 3 \lambda K_m^3 can be embedded in a one-factorization of λ K n 3 \lambda K_n^3 . In particular, we prove the first known Ryser type theorem for hypergraphs by showing that if n ≡ 0 ( m o d 3 ) n\equiv 0\ (\mathrm {mod}\ 3) , any edge-coloring of λ K m 3 \lambda K_m^3 where the number of triples of each color is at least m / 2 − n / 6 m/2-n/6 , can be embedded in a one-factorization of λ K n 3 \lambda K_n^3 . Finally we prove an Evans type result by showing that if n ≡ 0 ( m o d 3 ) n\equiv 0\ (\mathrm {mod}\ 3) and n ≥ 3 m n\geq 3m , then any q q -coloring of the edges of any F ⊆ λ K m 3 F\subseteq \lambda K_m^3 can be embedded in a one-factorization of λ K n 3 \lambda K_n^3 as long as q ≤ λ ( n − 1 2 ) − λ ( m 3 ) / ⌊ m / 3 ⌋ q\leq \lambda \binom {n-1}{2}-\lambda \binom {m}{3}/\left \lfloor m/3\right \rfloor .

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Non-extendible latin parallelepipeds

Cited by 2 publications

References 12 publications

Ryser Type Conditions for Extending Colorings of Triples

Ryser Type Conditions for Extending Colorings of Triples

Toward a three-dimensional counterpart of Cruse’s theorem

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