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Reed's omega, delta, and chi conjecture

Importance: High ✭✭✭
Author(s): Reed, Bruce A.
Subject: Graph Theory
» Coloring
» » Vertex coloring
Keywords: coloring
Recomm. for undergrads: no
Posted by: mdevos
on: May 22nd, 2007

For a graph $ G $, we define $ \Delta(G) $ to be the maximum degree, $ \omega(G) $ to be the size of the largest clique subgraph, and $ \chi(G) $ to be the chromatic number of $ G $.

Conjecture   $ \chi(G) \le \ceil{\frac{1}{2}(\Delta(G)+1) + \frac{1}{2}\omega(G)} $ for every graph $ G $.

Perhaps the two most trivial bounds on $ \chi(G) $ are $ \chi(G) \ge \omega(G) $ and $ \chi(G) \le \Delta(G) + 1 $. The above conjecture roughly asserts that the (rounded-up) average of $ \Delta(G)+1 $ and $ \omega(G) $ should again be an upper bound on $ \chi(G) $.

The conjecture is easy to verify when $ \omega(G) $ is very large. It is trivial when $ \omega(G) \ge \Delta(G) $, and it follows from Brook's theorem if $ \omega(G) = \Delta(G)-1 $. On the other hand, if $ \omega(G) = 2 $, so $ G $ is triangle free, then the conjecture is also true for $ \Delta $ sufficiently large. Indeed, Johannsen proved the much stronger fact that there exists a fixed constant $ c $ so that $ \chi(G) \le \frac{c \Delta(G)}{\log \Delta(G)} $ for every triangle free graph $ G $.

Reed showed that the conjecture holds when $ \Delta(G) = |V(G)| - 1 $ by way of matching theory. More interestingly, he proved (using probabilistc methods) that the conjecture is true provided that $ \Delta $ is sufficiently large, and $ \omega $ is sufficiently close to $ \Delta $. More precisely, he proves the following:

Theorem   There exists a fixed constant $ \Delta_0 $ such that for every $ \Delta \ge \Delta_0 $, if $ G $ is a graph of maximum degree $ \Delta $ with no clique of size $ >k $ for some $ k \ge (1 - \frac{1}{70000000}) \Delta $ then $ \chi(G) \le \frac{\Delta + 1 + k}{2} $.

It is known that the conjecture is true fractionally (that is with $ \chi(G) $ replaced by $ \chi_f(G) $, the fractional chromatic number of~$ G $).

Bibliography

*[R] B. Reed, $ \omega, \Delta $, and $ \chi $, J. Graph Theory 27 (1998) 177-212.


* indicates original appearance(s) of problem.
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The statement of the

On September 18th, 2007 Anonymous says:

The statement of the conjecture is slightly incorrect. Instead of the +1 at the end, there should simply be a round-up. The conjecture is true for line graphs and quasi-line graphs, graphs with independence number 2, and any graph on I believe 12 vertices.

An outright proof of the result for triangle-free graphs would be very nice. Lovasz' result on splitting graphs is not quite enough in this case.

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changed

On September 24th, 2007 Robert Samal says:

Thanks for the comment. I changed the statement to the conjectured (slightly stronger) version with round-up.

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