From arXiv

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 this it follows that the conjecture holds for graphs with $\chi(G) > \left \lceil \frac{|G|}{2} \right \rceil$. In addition, the conjecture holds for graphs with $\Delta(G) \geq |G| - \sqrt{|G| + 2\alpha(G) + 1}$. In particular, Reed's conjecture holds for graphs with $\Delta(G) \geq |G| - \sqrt{|G| + 7}$. Using these results, we proceed to show that if $|G|$ is an even order counterexample to Reed's conjecture, then $\bar{G}$ has a 1-factor. Hence, for any even order graph $G$, if $\chi(G) > \textstyle {1/2}(\omega(G) + \Delta(G) + 1) + 1$, then $\bar{G}$ is matching covered.

Simplify

Published on April 24, 2006

Copy BibTeX

Loading...

Summary

There is no AI-powered summary yet, because we do not have a budget to generate summaries for all articles.

1. Buy subscription

We will thank you for helping thousands of people to save their time at the top of the generated summary.

If you buy our subscription, you will be able to summarize multiple articles.

See an example

Pay $8

≈10 summaries

Pay $32

≈60 summaries

2. Share on socials

If this article gets to top-5 in trends, we'll summarize it for free.

Copy link