Related papers: Lower bounds for measurable chromatic numbers
The problems of computing graph colorings and clique covers are central challenges in combinatorial optimization. Both of these are known to be NP-hard, and thus computationally intractable in the worst-case instance. A prominent approach…
We prove lower and upper bounds for the chromatic number of certain hypergraphs defined by geometric regions. This problem has close relations to conflict-free colorings. One of the most interesting type of regions to consider for this…
We define the chromatic measure of a finite simple graph as the uniform distribution on its chromatic roots. We show that for a Benjamini-Schramm convergent sequence of finite graphs, the chromatic measures converge in holomorphic moments.…
In this paper, we take a modest first step towards a systematic study of chromatic numbers of Cayley graphs on abelian groups. We lose little when we consider these graphs only when they are connected and of finite degree. As in the work of…
In this paper, we obtain new asymptotic lower bounds for the chromatic numbers of spheres.
One of the best known results in spectral graph theory is the following lower bound on the chromatic number due to Alan Hoffman, where mu_1 and mu_n are respectively the maximum and minimum eigenvalues of the adjacency matrix: chi >= 1 +…
In this paper, we will study the chromatic number of Cayley graphs of algebraic groups that arise from algebraic constructions. Using Lang-Weil bound and representation theory of finite simple groups of Lie type, we will establish lower…
The dichromatic number $\vec{\chi}(D)$ of a digraph $D$ is the least number $k$ such that the vertex set of $D$ can be partitioned into $k$ parts each of which induces an acyclic subdigraph. Introduced by Neumann-Lara in 1982, this digraph…
The chromatic threshold $\delta_\chi(H)$ of a graph $H$ is the infimum of $d>0$ such that the chromatic number of every $n$-vertex $H$-free graph with minimum degree at least $dn$ is bounded in terms of $H$ and $d$. A breakthrough result of…
The study of very large graphs is a prominent theme in modern-day mathematics. In this paper we develop a rigorous foundation for studying the space of finite labelled graphs and their limits. These limiting objects are naturally countable…
By Lovasz' proof of the Kneser conjecture, the chromatic number of a graph G is bounded from below by the index of the Z_2-space Hom(K_2,G) plus two. We show that the cohomological index of Hom(K_2,G) is also greater than the cohomological…
For $q,n,d \in \mathbb{N}$, let $A_q^L(n,d)$ denote the maximum cardinality of a code $C \subseteq \mathbb{Z}_q^n$ with minimum Lee distance at least $d$, where $\mathbb{Z}_q$ denotes the cyclic group of order $q$. We consider a…
In this paper we prove a conjecture by Wocjan, Elphick and Anekstein (2018) which upper bounds the sum of the squares of the positive (or negative) eigenvalues of the adjacency matrix of a graph by an expression that behaves monotonically…
Let $\text{ch}(G)$ denote the choice number of a graph $G$ (also called "list chromatic number" or "choosability" of $G$). Noel, Reed, and Wu proved the conjecture of Ohba that $\text{ch}(G)=\chi(G)$ when $|V(G)|\le 2\chi(G)+1$. We extend…
The orbital bivariate chromatic polynomial, introduced in this article, counts the number of ways to color the vertices of a graph with $\lambda$ colors such that adjacent vertices either receive distinct colors from a set of $\lambda$…
We prove that the acyclic chromatic number of a graph with maximum degree $\Delta$ is less than $2.835\Delta^{4/3}+\Delta$. This improves the previous upper bound, which was $50\Delta^{4/3}$. To do so, we draw inspiration from works by…
We introduce a new graph invariant that measures fractional covering of a graph by cuts. Besides being interesting in its own right, it is useful for study of homomorphisms and tension-continuous mappings. We study the relations with…
We derive upper and lower bounds on the degree $d$ for which the Lov\'asz $\vartheta$ function, or equivalently sum-of-squares proofs with degree two, can refute the existence of a $k$-coloring in random regular graphs $G_{n,d}$. We show…
We present a new method for reducing the size of graphs with a given property. Our method, which is based on clausal proof minimization, allowed us to compute several 553-vertex unit-distance graphs with chromatic number 5, while the…
The quantum chromatic number, $\chi_q(G)$, of a graph $G$ was originally defined as the minimal number of colors necessary in a quantum protocol in which two provers that cannot communicate with each other but share an entangled state can…