Related papers: Colored interlacing triangles and Genocchi medians
The quantum color coding scheme proposed by Korff and Kempe (quant-ph/0405086) is easily extended so that the color coding quantum system is allowed to be entangled with an extra auxiliary quantum system. It is shown that in the extended…
The colored neighborhood metric for sparse graphs was introduced by Bollob\'as and Riordan. The corresponding convergence notion refines a convergence notion introduced by Benjamini and Schramm. We prove that even in this refined sense, the…
This paper discusses reformulations of the problem of coloring plane maps with four colors. The context is the edge-coloring with three colors of cubic graphs such that three distinct colors occur at each vertex. We include discussion of…
In this paper we introduce mixed coloured permutation, permutations with certain coloured cycles, and study the enumerative properties of these combinatorial objects. We derive the generating function, closed forms, recursions and…
A well-known theorem of Spencer shows that any set system with $n$ sets over $n$ elements admits a coloring of discrepancy $O(\sqrt{n})$. While the original proof was non-constructive, recent progress brought polynomial time algorithms by…
A proper $q$-coloring of a graph is an assignment of one of $q$ colors to each vertex of the graph so that adjacent vertices are colored differently. Sample uniformly among all proper $q$-colorings of a large discrete cube in the integer…
The colorful simplicial depth of a collection of d+1 finite sets of points in Euclidean d-space is the number of choices of a point from each set such that the origin is contained in their convex hull. We use methods from combinatorial…
We introduce a new model of random multigraphs with colored vertices and weighted edges. It is similar to the "inhomogeneous random graph model" of S\"oderberg (2002), extended by Bollob\'as, Janson and Riordan (2007). By means of analytic…
We present new combinatorial objects, which we call grid-labelled graphs, and show how these can be used to represent the quantum states arising in a scenario which we refer to as the faulty emitter scenario: we have a machine designed to…
A graph embedded in a surface with all faces of size 4 is known as a quadrangulation. We extend the definition of quadrangulation to higher dimensions, and prove that any graph G which embeds as a quadrangulation in the real projective…
This work is dedicated to $\mathfrak{sl}_{n+1}$-related integrable stochastic vertex models; we call such models coloured. We prove several results about these models, which include the following: (1) We construct the basis of (rational)…
Motivated by algorithmic applications, Kun, O'Brien, Pilipczuk, and Sullivan introduced the parameter linear chromatic number as a relaxation of treedepth and proved that the two parameters are polynomially related. They conjectured that…
Regular edge-colored graphs encode colored triangulations of pseudo-manifolds. Here we study families of edge-colored graphs built from a finite but arbitrary set of building blocks, which extend the notion of $p$-angulations to arbitrary…
Intersecting and cross-intersecting families usually appear in extremal combinatorics in the vein of the Erd{\H o}s--Ko--Rado theorem. On the other hand, P.~Erd{\H o}s and L.~Lov{\'a}sz in the noted paper~\cite{EL} posed problems on…
We study two parameters obtained from the Euler characteristic by replacing the number of faces with that of induced and induced non-separating cycles. By establishing monotonicity of such parameters under certain homomorphism and edge…
Given positive integers $n\ge 2k$, the {\it Kneser graph} $KG_{n,k}$ is a graph whose vertex set is the collection of all $k$-element subsets of the set $\{1,\ldots, n\}$, with edges connecting pairs of disjoint sets. One of the classical…
Random spatial networks-that is, graphs whose connectivity is governed by geometric proximity-have emerged as fundamental models for systems constrained by an underlying spatial structure. A prototypical example is the random geometric…
The renormalizable coloron model constitutes the minimal extension of the standard model (SM) color sector to $SU(3)_{1c} \times SU(3)_{2c}$, with the spontaneous symmetry breaking of the extended gauge group to the diagonal QCD facilitated…
Let $T(H, G_n)$ be the number of monochromatic copies of a fixed connected graph $H$ in a uniformly random coloring of the vertices of the graph $G_n$. In this paper we give a complete characterization of the limiting distribution of $T(H,…
In this paper we study threshold coloring of graphs, where the vertex colors represented by integers are used to describe any spanning subgraph of the given graph as follows. Pairs of vertices with near colors imply the edge between them is…