Related papers: Colouring Graphs with Sparse Neighbourhoods: Bound…

Let $G=(V,E)$ be a simple graph of maximum degree $\Delta$. The edges of $G$ can be colored with at most $\Delta +1$ colors by Vizing's theorem. We study lower bounds on the size of subgraphs of $G$ that can be colored with $\Delta$ colors.…

The chromatic number $\chi(G)$ of a graph $G$ is defined as the minimum number of colours required for a vertex colouring where no two adjacent vertices are coloured the same. The chromatic number of the dense random graph $G \sim G(n,p)$…

King, Lu, and Peng recently proved that for $\Delta\geq 4$, any $K_\Delta$-free graph with maximum degree $\Delta$ has fractional chromatic number at most $\Delta-\tfrac{2}{67}$ unless it is isomorphic to $C_5\boxtimes K_2$ or $C_8^2$.…

The chromatic edge stability index $\mathrm{es}_{\chi'}(G)$ of a graph $G$ is the minimum number of edges whose removal results in a graph with smaller chromatic index. We give best-possible upper bounds on $\mathrm{es}_{\chi'}(G)$ in terms…

Let $H$ be a graph with $\Delta(H) \leq 2$, and let $G$ be obtained from $H$ by gluing in vertex-disjoint copies of $K_4$. We prove that if $H$ contains at most one odd cycle of length exceeding $3$, or if $H$ contains at most $3$…

In 1977, Borodin and Kostochka conjectured that every graph with maximum degree $\Delta \geq 9$ is $(\Delta-1)$-colourable, unless it contains a clique of size $\Delta$. In 1999, Reed confirmed the conjecture when $\Delta\geq 10^{14}$. We…

The {\em square} $G^2$ of a graph $G$ is the graph with the same vertex set as $G$ and with two vertices adjacent if their distance in $G$ is at most 2. Thomassen showed that every planar graph $G$ with maximum degree $\Delta(G)=3$…

An $r$-hued coloring of a simple graph $G$ is a proper coloring of its vertices such that every vertex $v$ is adjacent to at least $\min\{r, \deg(v)\}$ differently colored vertices. The minimum number of colors needed for an $r$-hued…

Let $\Delta(G)$ be the maximum degree of a graph $G$. Brooks' theorem states that the only connected graphs with chromatic number $\chi(G)=\Delta(G)+1$ are complete graphs and odd cycles. We prove a fractional analogue of Brooks' theorem in…

The \emph{choice number} of a graph $G$, denoted $\ch(G)$, is the minimum integer $k$ such that for any assignment of lists of size $k$ to the vertices of $G$, there is a proper colouring of $G$ such that every vertex is mapped to a colour…

The closed neighborhood conflict-free chromatic number of a graph $G$, denoted by $\chi_{CN}(G)$, is the minimum number of colors required to color the vertices of $G$ such that for every vertex, there is a color that appears exactly once…

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…

Let $A\subset\mathbb{R}_{>0}$ be a finite set of distances, and let $G_{A}(\mathbb{R}^{n})$ be the graph with vertex set $\mathbb{R}^{n}$ and edge set $\{(x,y)\in\mathbb{R}^{n}:\ \|x-y\|_{2}\in A\}$, and let…

Given independent random points $X_1,...,X_n\in\eR^d$ with common probability distribution $\nu$, and a positive distance $r=r(n)>0$, we construct a random geometric graph $G_n$ with vertex set $\{1,...,n\}$ where distinct $i$ and $j$ are…

The {\em acyclic chromatic number} of a graph is the least number of colors needed to properly color its vertices so that none of its cycles has only two colors. The {\em acyclic chromatic index} is the analogous graph parameter for edge…

Given a graph $G$, denote by $\Delta$, $\bar{d}$ and $\chi^\prime$ the maximum degree, the average degree and the chromatic index of $G$, respectively. A simple graph $G$ is called {\it edge-$\Delta$-critical} if $\chi^\prime(G)=\Delta+1$…

Let $G$ be a simple graph with maximum degree $\Delta(G)$ and chromatic index $\chi'(G)$. A classic result of Vizing indicates that either $\chi'(G )=\Delta(G)$ or $\chi'(G )=\Delta(G)+1$. The graph $G$ is called $\Delta$-critical if $G$ is…

We show that for a simple graph $G$, $c'(G)\leq\Delta(G)+2$ where $c'(G)$ is the choice index (or edge-list chromatic number) of $G$, and $\Delta(G)$ is the maximum degree of $G$. As a simple corollary of this result, we show that the total…

Given a graph or multigraph $G$, let $\chi'_{trans}(G)$ denote the minimum integer $n$ such that any proper $\chi'(G)$--edge coloring of $G$ can be transformed into any other proper $\chi'(G)$--edge coloring of $G$ by a series of…

We prove that if $G$ is a vertex critical graph with $\chi(G) \geq \Delta(G) + 1 - p \geq 4$ for some $p \in \mathbb{N}$ and $\omega(\fancy{H}(G)) \leq \frac{\chi(G) + 1}{p + 1} - 2$, then $G = K_{\chi(G)}$ or $G = O_5$. Here $\fancy{H}(G)$…