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From arXiv

A note on Reed's conjecture

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

\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

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