Related papers: Chromatic Number of Grassmann Graphs and MRD codes

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…

We show that every (sub)cubic n-vertex graph with sufficiently large girth has fractional chromatic number at most 2.2978 which implies that it contains an independent set of size at least 0.4352n. Our bound on the independence number is…

A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most 1. The equitable chromatic number of a graph $G$, denoted by $\chi_=(G)$, is the minimum $k$ such that $G$ is equitably $k$-colorable. The…

We present an analysis of the structure and properties of chromatic polynomials $P(G_{pt,\vec m},q)$ of one-parameter and multi-parameter families of planar triangulation graphs $G_{pt,\vec m}$, where ${\vec m} = (m_1,...,m_p)$ is a vector…

The chromatic threshold of a graph $H$ is the minimum-degree density above which every $H$-free graph has bounded chromatic number. We study a two-color Ramsey analogue: for graphs $H_1$ and $H_2$, we ask for the minimum-degree density…

Let $n, r, k$ be positive integers such that $3\leq k < n$ and $2\leq r \leq k-1$. Let $m(n, r, k)$ denote the maximum number of edges an $r$-uniform hypergraph on $n$ vertices can have under the condition that any collection of $i$ edges,…

For integers $k\ge 1$ and $m\ge 2$, let $g(k,m)$ be the least integer $n\ge 1$ such that every graph with chromatic number at least $n$ contains a $(k+1)$-connected subgraph with chromatic number at least $m$. We prove that \[ g(k,m)\le…

For a simple graph $G$, let $\chi_f(G)$ be the fractional chromatic number of $G$. In this paper, we aim to establish upper bounds on $\chi_f(G)$ for those graphs $G$ with restrictions on the clique number. Namely, we prove that for $\Delta…

A graph is called a pseudo-core if every endomorphism is either an automorphism or a colouring. In this paper, we show that every Grassmann graph $J_q(n,m)$ is a pseudo-core. Moreover, the Grassmann graph $J_q(n,m)$ is a core whenever $m$…

For $d > 0$, define $G(\mathbb{Q}^3, d)$ to be the graph whose set of vertices is the rational space $\mathbb{Q}^3$, where two vertices are adjacent if and only if they are a Euclidean distance $d$ apart. Let $\chi(\mathbb{Q}^3, d)$ be the…

A coloring is called $s$-wide if no walk of length $2s-1$ connects vertices of the same color. A graph is $s$-widely colorable with $t$ colors if and only if it admits a homomorphism into a universal graph $W(s,t)$. Tardif observed that the…

The chromatic number $\chi$ of a graph is bounded from below by its clique number $\omega,$ but it can be arbitrary large. Perfect graphs are defined by $\chi=\omega$ for all induced subgraphs. An interesting relaxation are $\chi$-bounded…

We compute the exact value of the quantum chromatic numbers of Hadamard graphs of order $n=2^N$ for $N$ a multiple of $4$ using the upper bound derived by Avis, Hasegawa, Kikuchi, and Sasaki, as well as an application of the Hoffman-like…

Let $G$ be a graph each component of which has order at least 3, and let $G$ have order $n$, size $m$, total domination number $\gamma_t$ and maximum degree $\Delta(G)$. Let $\Delta = 3$ if $\Delta(G) = 2$ and $\Delta = \Delta (G)$ if…

A proper vertex colouring of a graph is \emph{nested} if the vertices of each of its colour classes can be ordered by inclusion of their open neighbourhoods. Through a relation to partially ordered sets, we show that the nested chromatic…

A Nordhaus-Gaddum-type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. In this paper some variations are considered. First, recall their theorem, which gives bounds on the sum and…

Let G=(V,E) be a graph of order n without isolated vertices. A bijection f:V -- {1,2,...n} is called a local distance antimagic labeling if the weights of any two adjacent vertices are not equal, where the weight of a vertex is defined to…

We prove that for an arbitrary $\varepsilon > 0$ holds \[ \chi (\mathbb{R}^3 \times [0,\varepsilon]^6) \geq 10, \] where $\chi(M)$ stands for the chromatic number of an (infinite) graph with the vertex set $M$ and the edge set consists of…

The sphere graph of $M_r$, a connect sum of $r$ copies of $S^1\times S^2$ was introduced by Hatcher as an analog of the curve graph of a surface to study the outer automorphism group of a free group $F_r$. Bestvina, Bromberg, and Fujiwara…

The acyclic chromatic index (or acyclic edge-chromatic number) of a graph is the least number of colors needed to properly color its edges so that none of its cycles has only two colors. We show that for a graph of max degree $\Delta$, the…