The Wide Partition Conjecture and the Atom Problem in Discrete Tomography

Dürr, Christoph; Guiñez, Flavio

doi:10.1016/j.endm.2013.10.055

Cited by 2 publications

(5 citation statements)

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“…However, by modifying our arguments slightly, we can show that G (λ) has clique number four in this case as well. s (3,4) or c (3,4) s (3,4) Proof. Let T be a filling of λ, and consider a triangle C in G (T ).…”

Section: Clique Numbers Of Isotopy Graphsmentioning

confidence: 98%

“…We have seen that the clique number of G (λ) is 4 when λ = (4, 4). Suppose now that λ is not equal to (4,4).…”

Section: Clique Numbers Of Isotopy Graphsmentioning

confidence: 99%

“…Let T be a Latin tableaux of shape λ. Then G (T ) has a triangle if and only if there exist elementary transformations r = r (i1,i2) ∈ S row , c = c (j1,j2) ∈ S col and s = s (a1,a2) ∈ S ent such that a 1 , a 2 ∈ [4], rc(T ) = s(T ) and all of the three following properties hold:…”

Section: Clique Numbers Of Isotopy Graphsmentioning

confidence: 99%

“…However, the transformation c (1,2) , which switches the first two columns of T , acts identically to s (1,2) , which switches entries 1 and 2. Similarly c (3,4) , the transformation which switches the columns 3 and 4, acts identically to s (3,4) , which switches entries 3 and 4. Hence there are only 11 elementary transformations which act distinctly on T , and the degree of S (T ) is 11.…”

Section: Degrees Of Isotopy Graphsmentioning

confidence: 99%

“…The conjecture may be re-stated in a number of contexts, including list colorings and network flows [2]. Dürr and Guíñez gave a reformulation of the WPC from the perspective discrete tomography, the study of reconstructing geometric objects from their projections to finite sets of integers [4]. More specifically, they translated the Wide Partition Conjecture into a claim about whether certain combinations of integer vectors can be realized as projections of finite subsets of a discrete lattice.…”

Section: Introductionmentioning

confidence: 99%

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Isotopy graphs of Latin tableaux

Karpman

Roldán

2020

Preprint

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Latin tableaux are a generalization of Latin squares, which first appeared in the early 2000's in a paper of Chow, Fan, Goemans, and Vondrák. Here, we extend the notion of isotopy, a permutation group action, from Latin squares to Latin tableaux. We define isotopy graphs for Latin tableaux, which encode the structure of orbits under the isotopy action, and investigate the relationship between the shape of a Latin tableau and the structure of its isotopy graph. Our main result shows that for any positive integer d, there is a Latin tableau whose isotopy graph is a d-dimensional cube. We show that most isotopy graphs are triangle-free, and we give a characterization of all the Latin tableaux for which the isotopy graph contains a triangle. We also give a formula for the degree of each component of an isotopy graph, which depends on both the shape of the Latin Tableaux and the filling.

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