相关论文: The last fraction of a fractional conjecture

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…

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, 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…

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.…

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 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's well-known $\omega$, $\Delta$, $\chi$ conjecture proposes that every graph satisfies $\chi \leq \lceil \frac 12(\Delta+1+\omega)\rceil$. The second author formulated a {\em local strengthening} of this conjecture that considers a…

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 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…

We introduce a new method for computing bounds on the independence number and fractional chromatic number of classes of graphs with local constraints, and apply this method in various scenarios. We establish a formula that generates a…

The dichromatic number of a graph $G$ is the maximum integer $k$ such that there exists an orientation of the edges of $G$ such that for every partition of the vertices into fewer than $k$ parts, at least one of the parts must contain a…

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…

Given a graph $G$ possibly with multiple edges but no loops, denote by $\Delta$ the {\it maximum degree}, $\mu$ the {\it multiplicity}, $\chi'$ the {\it chromatic index} and $\chi_f'$ the {\it fractional chromatic index} of $G$,…

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 $C_8^2$.…

Let $G$ be a simple graph with maximum degree $\Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>\Delta(G)\lfloor |V(H)|/2 \rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ with $\Delta(G)>|V(G)|/3$ has chromatic…

We prove that for all $\varepsilon>0$, there exists a positive integer $n_0$ such that if $G$ is a graph on $n\geq n_0$ vertices with $\delta(G)\geq\tfrac{1}{2}(1 + \varepsilon)n$, then $G$ satisfies the Total Coloring Conjecture, that is,…

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…

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…

Given $\varepsilon>0$, there exists $f_0$ such that, if $f_0 \le f \le \Delta^2+1$, then for any graph $G$ on $n$ vertices of maximum degree $\Delta$ in which the neighbourhood of every vertex in $G$ spans at most $\Delta^2/f$ edges, (i) an…