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A note on Reed's conjecture

In \cite{reed97}, Reed conjectures that the inequality χ(G)1/2(ω(G)+Δ(G)+1)\chi (G) \leq \left \lceil \textstyle {1/2} (\omega (G) + \Delta (G) + 1) \right \rceil holds for any graph GG. We prove this holds for a graph GG if Gˉ\bar{G} is disconnected. From this it follows that the conjecture holds for graphs with χ(G)>G2\chi(G) > \left \lceil \frac{|G|}{2} \right \rceil. In addition, the conjecture holds for graphs with Δ(G)GG+2α(G)+1\Delta(G) \geq |G| - \sqrt{|G| + 2\alpha(G) + 1}. In particular, Reed's conjecture holds for graphs with Δ(G)GG+7\Delta(G) \geq |G| - \sqrt{|G| + 7}. Using these results, we proceed to show that if G|G| is an even order counterexample to Reed's conjecture, then Gˉ\bar{G} has a 1-factor. Hence, for any even order graph GG, if χ(G)>1/2(ω(G)+Δ(G)+1)+1\chi(G) > \textstyle {1/2}(\omega(G) + \Delta(G) + 1) + 1, then Gˉ\bar{G} is matching covered.
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Published on April 24, 2006
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