Related papers: On Cobweb Posets and Discrete F-Boxes Tilings
Partially ordered sets labeled with k labels (k-posets) and their homomorphisms are examined. We give a representation of directed graphs by k-posets; this provides a new proof of the universality of the homomorphism order of k-posets. This…
Subgraph counting is a fundamental and well-studied problem whose computational complexity is well understood. Quite surprisingly, the hypergraph version of subgraph counting has been almost ignored. In this work, we address this gap by…
Trellises are crucial graphical representations of codes. While conventional trellises are well understood, the general theory of (tail-biting) trellises is still under development. Iterative decoding concretely motivates such theory. In…
For a graph $H$, a graph $G$ is an $H$-graph if it is an intersection graph of connected subgraphs of some subdivision of $H$. $H$-graphs naturally generalize several important graph classes like interval or circular-arc graph. This class…
It is known that a number of natural graph problems which are FPT parameterized by treewidth become W-hard when parameterized by clique-width. It is therefore desirable to find a different structural graph parameter which is as general as…
We study the geometry of elliptic fibrations satisfying the conditions of Step 8 of Tate's algorithm. We call such geometries F$_4$-models, as the dual graph of their special fiber is the twisted affine Dynkin diagram…
Grothendieck's theory of dessins provides a bridge between algebraic numbers and combinatorics. This paper adds a new concept, called 'bias', to the bridge. This produces: (i) from a biased plane tree the construction of a sequence of…
Box graphs, or equivalently Coulomb phases of three-dimensional N=2 supersymmetric gauge theories with matter, give a succinct, comprehensive and elegant characterization of crepant resolutions of singular elliptically fibered varieties.…
This thesis establishes a generalised setting with which to unify the study of finite local complexity (FLC) patterns. The abstract notion of a "pattern" is introduced, which may be seen as an analogue of the space group of isometries…
A wide variety of problems in combinatorics and discrete optimization depend on counting the set $S$ of integer points in a polytope, or in some more general object constructed via discrete geometry and first-order logic. We take a tour…
We consider the Minimum Coverage Kernel problem: given a set $B$ of $d$-dimensional boxes, find a subset of $B$ of minimum size covering the same region as $B$. This problem is $\mathsf{NP}$-hard, but as for many $\mathsf{NP}$-hard problems…
We solve the existence problem for $F$-designs for arbitrary $r$-uniform hypergraphs~$F$. This implies that given any $r$-uniform hypergraph~$F$, the trivially necessary divisibility conditions are sufficient to guarantee a decomposition of…
We compute the diagonal F-thresholds of determinantal hypersurfaces arising from a generic matrix and from a generic symmetric matrix, as well as of the Pfaffian hypersurface arising from a generic skew-symmetric matrix of even size. The…
A multifamily set representation of a finite simple graph $G$ is a multifamily $\mathcal{F}$ of sets (not necessarily distinct) for which each set represents a vertex in $G$ and two sets in $\mathcal{F}$ intersects if and only if the two…
We show that an effective version of Siegel's Theorem on finiteness of integer solutions and an application of elementary Galois theory are key ingredients in a complexity classification of some Holant problems. These Holant problems,…
We examine the proposal in the previous paper to resolve the puzzle in transitions in discrete gauge groups. We focus on a four-section geometry to test the proposal. We observed that a discrete $\mathbb{Z}_2$ gauge group enlarges and…
In arXiv:1405.4527 Connes and Consani introduced and studied the arithmetic site and showed that the isomorphism classes of points are in canonical bijection with the finite adele classes $\mathbb{Q}^*_+ \backslash \mathbb{A}^f_{\mathbb{Q}}…
For each positive integer $n$, let $G_n$ be the graph of integer partitions of $n$, where two partitions are adjacent if one is obtained from the other by an elementary transfer of a cell in the Ferrers diagram, followed by reordering.…
Fractional (hyper-)graph theory is concerned with the specific problems that arise when fractional analogues of otherwise integer-valued (hyper-)graph invariants are considered. The focus of this paper is on fractional edge covers of…
Partitioning large matrices is an important problem in distributed linear algebra computing (used in ML among others). Briefly, our goal is to perform a sequence of matrix algebra operations in a distributed manner (whenever possible) on…