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Solving a conjecture of M. H. Freedman, L. Lov\'asz and A. Schrijver we prove that a graph parameter is edge reflection positive and multiplicative if and only if it can be represented by an edge coloring model.
This paper introduces a notion of decompositions of integral varifolds into countably many integral varifolds, and the existence of such decomposition of integral varifolds whose first variation is representable by integration is…
A graph is IC-planar if it admits a drawing in the plane with at most one crossing per edge and such that two pairs of crossing edges share no common end vertex. IC-planarity specializes both NIC-planarity, which allows a pair of crossing…
We uncover a connection between two seemingly unrelated notions: lettericity, from structural graph theory, and geometric griddability, from the world of permutation patterns. Both of these notions capture important structural properties of…
We provide a characterization of infinite frieze patterns of positive integers via triangulations of an infinite strip in the plane. In the periodic case, these triangulations may be considered as triangulations of annuli. We also give a…
We introduce the notion of recurrence and transience for graphs over non-Archimedean ordered field. To do so we relate these graphs to random walks of directed graphs over the reals. In particular, we give a characterization of the real…
In this paper, we investigate the edge-coloring number of the power graph of a finite group. We characterize which finite groups have overfull power graphs, showing that this occurs if and only if the group is cyclic of odd prime power…
A vertex colouring of a graph is \emph{nonrepetitive} if there is no path whose first half receives the same sequence of colours as the second half. A graph is nonrepetitively $k$-choosable if given lists of at least $k$ colours at each…
In this study, we undertake a reproducibility analysis of 'Learning Fair Graph Representations Via Automated Data Augmentations' by Ling et al. (2022). We assess the validity of the original claims focused on node classification tasks and…
Permutation actions of simple currents on the primaries of a Rational Conformal Field Theory are considered in the framework of admissible weighted permutation actions. The solution of admissibility conditions is presented for cyclic…
We introduce the notion of porous invariants for multipath (or branching/nondeterministic) affine loops over the integers; these invariants are not necessarily convex, and can in fact contain infinitely many 'holes'. Nevertheless, we show…
We construct representation theory of Lie algebras with filtrations. In this framework a classification of irreducible representations is obtained and spectra of some reducible representations are found.
In this article we study the notion of capacity of a vertex for infinite graphs over non-Archimedean fields. In contrast to graphs over the real field monotone limits do not need to exist. Thus, in our situation next to positive and null…
This is an expository paper about iterations of a smooth real function $f$ on $[0,\varepsilon)$ such that $f(0)=0$, $f'(0)=1$, and $f(x)<x$ for $x>0$, i.e., the sequence defined by $x_{n+1}=f(x_n)$. This sequence has interesting…
If $(X,J)$ is an almost complex manifold, then a function $u$ is said to be plurisubharmonic on $X$ if it is upper semi-continuous and its restriction to every local pseudo-holomorphic curve is subharmonic. As in the complex case, it is…
A colouring of a graph is "nonrepetitive" if for every path of even order, the sequence of colours on the first half of the path is different from the sequence of colours on the second half. We show that planar graphs have nonrepetitive…
We review extensions by integer spin simple currents in two-dimensional conformal field theories and their applications in string theory. In particular, we study the problem of resolving the fixed points of a simple current and apply the…
In this paper we define currents relative to a free factor system. We prove that a fully irreducible outer automorphism relative to a free factor system acts with uniform north-south dynamics on a subspace of the space of projective…
We investigate spectral properties of quantum graphs in the form of a periodic chain of rings with a connecting link between each adjacent pair, assuming that wave functions at the vertices are matched through conditions manifestly…
In any vertex coloring of a graph some edges have differently colored ends (\emph{good} edges) and some are monochromatic (\emph{bad} edges). In a proper coloring all edges are good. In a \emph{majority coloring} it is enough that for every…