Related papers: A superlocal version of Reed's Conjecture

Reed's $\omega$, $\Delta$, $\chi$ conjecture proposes that every graph satisfies $\chi\leq \lceil\frac 12(\Delta+1+\omega)\rceil$; it is known to hold for all claw-free graphs. In this paper we consider a local strengthening of this…

Let $G$ be a graph with chromatic number $\chi$, maximum degree $\Delta$ and clique number $\omega$. Reed's conjecture states that $\chi \leq \lceil (1-\varepsilon)(\Delta + 1) + \varepsilon\omega \rceil$ for all $\varepsilon \leq 1/2$. It…

Reed conjectured that for every graph, $\chi \leq \left \lceil \frac{\Delta + \omega + 1}{2} \right \rceil$ holds, where $\chi$, $\omega$ and $\Delta$ denote the chromatic number, clique number and maximum degree of the graph, respectively.…

In 1998, Reed conjectured that every graph $G$ satisfies $\chi(G) \leq \lceil \frac{1}{2}(\Delta(G) + 1 + \omega(G))\rceil$, where $\chi(G)$ is the chromatic number of $G$, $\Delta(G)$ is the maximum degree of $G$, and $\omega(G)$ is the…

Reed conjectured that for any graph $G$, $\chi(G) \leq \lceil \frac{\omega(G)+\Delta(G)+1}{2}\rceil$, where $\chi(G)$, $\omega(G)$, and $\Delta(G)$ respectively denote the chromatic number, the clique number and the maximum degree of $G$.…

In \cite{reed97}, Reed conjectures that the inequality $\chi (G) \leq \left \lceil \textstyle {1/2} (\omega (G) + \Delta (G) + 1) \right \rceil$ holds for any graph $G$. We prove this holds for a graph $G$ if $\bar{G}$ is disconnected. From…

In 1998, Reed conjectured that for any graph $G$, $\chi(G) \leq \lceil \frac{\omega(G) + \Delta(G)+1}{2}\rceil$, where $\chi(G)$, $\omega(G)$, and $\Delta(G)$ respectively denote the chromatic number, the clique number and the maximum…

Reed conjectured that for every $\varepsilon>0$ and every integer $\Delta$, there exists $g$ such that the fractional total chromatic number of every graph with maximum degree $\Delta$ and girth at least $g$ is at most…

The second author's $\omega$, $\Delta$, $\chi$ conjecture proposes that every graph satisties $\chi \leq \lceil \frac 12 (\Delta+1+\omega)\rceil$. In this paper we prove that the conjecture holds for all claw-free graphs. Our approach uses…

We improve upper bounds on the chromatic number proven independently in \cite{reedNote} and \cite{ingo}. Our main lemma gives a sufficient condition for two paths in graph to be completely joined. Using this, we prove that if a graph has an…

We outline some ongoing work related to a conjecture of Reed \cite{reed97} on $\omega$, $\Delta$, and $\chi$. We conjecture that the complement of a counterexample $G$ to Reed's conjecture has connectivity on the order of $\log(|G|)$. We…

Reed in 1998 conjectured that every graph $G$ satisfies $\chi(G) \leq \lceil \frac{\Delta(G)+1+\omega(G)}{2} \rceil$. As a partial result, he proved the existence of $\varepsilon > 0$ for which every graph $G$ satisfies $\chi(G) \leq \lceil…

We develop an improved bound for the chromatic number of graphs of maximum degree $\Delta$ under the assumption that the number of edges spanning any neighbourhood is at most $(1-\sigma)\binom{\Delta}{2}$ for some fixed $0<\sigma<1$. The…

Reed conjectured that the chromatic number of any graph is closer to its clique number than to its maximum degree plus one. We consider a recolouring version of this conjecture, with respect to Kempe changes. Namely, we investigate the…

In 1998 the second author proved that there is an $\epsilon>0$ such that every graph satisfies $\chi \leq \lceil (1-\epsilon)(\Delta+1)+\epsilon\omega\rceil$. The first author recently proved that any graph satisfying $\omega > \frac…

Reed conjectured that for every epsilon>0 and Delta there exists g such that the fractional total chromatic number of a graph with maximum degree Delta and girth at least g is at most Delta+1+epsilon. We prove the conjecture for Delta=3 and…

We prove that every graph $G$ for which $\omega(G) \geq 3/4(\Delta(G) + 1)$, has an independent set $I$ such that $\omega(G - I) < \omega(G)$. It follows that a minimum counterexample $G$ to Reed's conjecture satisfies $\omega(G) <…

Given a graphic degree sequence $D$, let $\chi(D)$ (respectively $\omega(D)$, $h(D)$, and $H(D)$) denote the maximum value of the chromatic number (respectively, the size of the largest clique, largest clique subdivision, and largest clique…

We prove bounds on the chromatic number $\chi$ of a vertex-transitive graph in terms of its clique number $\omega$ and maximum degree $\Delta$. We conjecture that every vertex-transitive graph satisfies $\chi \le \max \left\{\omega,…

We give a short proof of a bound on the list chromatic number of graphs $G$ of maximum degree $\Delta$ where each neighbourhood has density at most $d$, namely $\chi_\ell(G) \le (1+o(1)) \frac{\Delta}{\ln \frac{\Delta}{d+1}}$ as…