Related papers: Young tableau reconstruction via minors
Beineke, Harary and Ringel discovered a formula for the minimum genus of a torus in which the $n$-dimensional hypercube graph can be embedded. We give a new proof of the formula by building this surface as a union of certain faces in the…
Minimizing the nuclear norm of a matrix has been shown to be very efficient in reconstructing a low-rank sampled matrix. Furthermore, minimizing the sum of nuclear norms of matricizations of a tensor has been shown to be very efficient in…
Discrete tomography deals with reconstructing finite spatial objects from lower dimensional projections and has applications for example in timetable design. In this paper we consider the problem of reconstructing a tile packing from its…
Pseudoline arrangements are fundamental objects in discrete and computational geometry, and different works have tackled the problem of improving the known bounds on the number of simple arrangements of $n$ pseudolines over the past…
To facilitate a simultaneous treatment of an arbitrary number of colors in representation theory-based descriptions of QCD color structure, we derive an $N$-independent reduction of SU($N$) tensor products. To this end, we label each…
In this paper, we study the functional introduced by the author in collaboration with Bonnivard, Bretin, and Lemenant, which is designed to approximate Plateau's problem. We establish the existence of a minimizer and prove its H{\"o}lder…
The Reconstruction Conjecture of Kelly and Ulam states that any graph $G$ with $n\geq 3$ vertices can be reconstructed from the multiset $\mathcal{D}(G)$ of unlabelled subgraphs $G-v$ for all $v\in V(G)$. We refer to $\mathcal{D}(G)$ as the…
We prove that for $n>k\geq 3$, if $G$ is an $n$-vertex graph with chromatic number $k$ but any its proper subgraph has smaller chromatic number, then $G$ contains at most $n-k+3$ copies of cliques of size $k-1$. This answers a problem of…
In 1963, Halin and Jung proved that every simple graph with minimum degree at least four has $K_5$ or $K_{2,2,2}$ as a minor. Mills and Turner proved an analog of this theorem by showing that every $3$-connected binary matroid in which…
Young tableaux are ubiquitous in various branches of mathematics. There are two counting formulas for standard Young tableaux. The first involves a determinant and goes back to Frobenius and Young, and the second is the hook formula by…
For integers $k\ge 2$ and $N\ge 2k+1$ there are $k!2^k$ canonical orderings of the edges of the complete $k$-uniform hypergraph with vertex set $[N] = \{1,2,\dots, N\}$. These are exactly the orderings with the property that any two subsets…
Let $G$ be an undirected, bounded degree graph with $n$ vertices. Fix a finite graph $H$, and suppose one must remove $\varepsilon n$ edges from $G$ to make it $H$-minor free (for some small constant $\varepsilon > 0$). We give an…
A fundamental result in structural graph theory states that every graph with large average degree contains a large complete graph as a minor. We prove this result with the extra property that the minor is small with respect to the order of…
An $(n, k, d, \alpha)$-MSR (minimum storage regeneration) code is a set of $n$ nodes used to store a file. For a file of total size $k\alpha$, each node stores $\alpha$ symbols, any $k$ nodes recover the file, and any $d$ nodes can repair…
In this paper we raise a variant of a classic problem in extremal graph theory, which is motivated by a design of fractional repetition codes, a model in distributed storage systems. For any feasible positive integers $d\geq 3$, $n \geq 3$,…
A matrix is called totally positive if every minor of it is positive. Such matrices are well studied and have numerous applications in Mathematics and Computer Science. We study how many times the value of a minor can repeat in a totally…
We prove that in every cover of a Young diagram with $\binom{2k}{k}$ steps with generalized rectangles there is a row or a column in the diagram that is used by at least $k+1$ rectangles. We show that this is best-possible by partitioning…
We present a method for reducing the treewidth of a graph while preserving all of its minimal $s-t$ separators up to a certain fixed size $k$. This technique allows us to solve $s-t$ Cut and Multicut problems with various additional…
We present the plactic algebra on an arbitrary alphabet set $A$ by row generators and column generators respectively. We give Gr\"{o}bner-Shirshov bases for such presentations. In the case of column generators, a finite Gr\"{o}bner-Shirshov…
We study the problem of determining the minimum number $f(n,k,d)$ of affine subspaces of codimension $d$ that are required to cover all points of $\mathbb{F}_2^n\setminus \{\vec{0}\}$ at least $k$ times while covering the origin at most…