Related papers: On the Quantum Chromatic Gap
We study a model of random graph where vertices are $n$ i.i.d. uniform random points on the unit sphere $S^d$ in $\mathbb{R}^{d+1}$, and a pair of vertices is connected if the Euclidean distance between them is at least $2- \epsilon$. We…
This paper is a study of ``topological'' lower bounds for the chromatic number of a graph. Such a lower bound was first introduced by Lov\'asz in 1978, in his famous proof of the \emph{Kneser conjecture} via Algebraic Topology. This…
We show that the Consensus Division theorem implies lower bounds on the chromatic number of Kneser hypergraphs, offering a novel proof for a result of Alon, Frankl, and Lov\'{a}sz (Trans. Amer. Math. Soc., 1986) and for its generalization…
We present an infinite sequence of finite graphs with trivial automorphism group and non-trivial quantum automorphism group. These are the first known examples of graphs with this property. Moreover, to the best of our knowledge, these are…
Consider the graph $\mathbb{H}(d)$ whose vertex set is the hyperbolic plane, where two points are connected with an edge when their distance is equal to some $d>0$. Asking for the chromatic number of this graph is the hyperbolic analogue to…
There are few explicit examples of two player nonlocal games with a large gap between classical and quantum value. One of the reasons is that estimating the classical value is usually a hard computational task. This paper is devoted to…
The Cyclic Coloring Conjecture asserts that the vertices of every plane graph with maximum face size D can be colored using at most 3D/2 colors in such a way that no face is incident with two vertices of the same color. The Cyclic Coloring…
Coloring a graph $G$ consists in finding an assignment of colors $c: V(G)\to\{1,\ldots,p\}$ such that any pair of adjacent vertices receives different colors. The minimum integer $p$ such that a coloring exists is called the chromatic…
Colored tensor models (CTM) is a random geometrical approach to quantum gravity. We scrutinize the structure of the connected correlation functions of general CTM-interactions and organize them by boundaries of Feynman graphs. For rank-$D$…
The dichromatic number of a digraph $D$ is the smallest $k$ such that $D$ can be partitioned into $k$ acyclic subdigraphs, and the dichromatic number of an undirected graph is the maximum dichromatic number over all its orientations.…
We prove that for any graph $G$, the total chromatic number of $G$ is at most $\Delta(G)+2\left\lceil \frac{|V(G)|}{\Delta(G)+1} \right\rceil$. This saves one color in comparison with a result of Hind from 1992. In particular, our result…
One of the key unsolved conjectures in hypergraph coloring is about the chromatic number of $s$-stable $r$-uniform Kneser hypergraphs $\mathrm{KG}^r(n,k)_{s\textup{-stab}}$. The problem remains largely open, particularly in the case where…
We develop a heuristic graph coloring approximation algorithm that uses the D-Wave 2X as an independent set sampler and evaluate its performance against a fully classical implementation. A randomly generated set of small but hard graph…
Gyarfas conjectured in 1985 that for all $k$, $l$, every graph with no clique of size more than $k$ and no odd hole of length more than $l$ has chromatic number bounded by a function of $k$ and $l$. We prove three weaker statements: (1)…
In their celebrated paper (arXiv:0906.1238), Caputo, Liggett and Richthammer proved Aldous' conjecture and showed that for an arbitrary finite graph, the spectral gap of the interchange process is equal to the spectral gap of the underlying…
The Additive Coloring Problem is a variation of the Coloring Problem where labels of $\{1,\ldots,k\}$ are assigned to the vertices of a graph $G$ so that the sum of labels over the neighborhood of each vertex is a proper coloring of $G$.…
A {\it simple $k$-coloring} of a multigraph $G$ is a decomposition of the edge multiset as a disjoint sum of $k$ simple graphs which are referred as colors. A subgraph $H$ of a multigraph $G$ is called {\it multicolored} if its edges…
Motivated from the theory of quantum error correcting codes, we investigate a combinatorial problem that involves a symmetric $n$-vertices colourable graph and a group of operations (colouring rules) on the graph: find the minimum sequence…
In this paper, in view of $Z_p$-Tucker lemma, we introduce a lower bound for chromatic number of Kneser hypergraphs which improves Dol'nikov-K{\v{r}}{\'{\i}}{\v{z}} bound. Next, we introduce multiple Kneser hypergraphs and we specify the…
\textit{Total Coloring} of a graph is a major coloring problem in combinatorial mathematics, introduced in the early $1960$s. A \textit{total coloring} of a graph $G$ is a map $f:V(G) \cup E(G) \rightarrow \mathcal{K}$, where $\mathcal{K}$…