Chromatic Number Upper Bound Conjectures

In our continued testing of the program we generated some conjectures for upper bounds for the chromatic number. We told the program the upper bounds due to Brooks, Wilf, and Welsh-Powell – so the program can’t make a conjecture implied by any of these. These were tested for the few dozen graphs the program knows and then some number of small-ish graphs.

The program seems to have noticed there is a relations chip between the chromatic and clique numbers of a graph: the clique number shows up in a large number of the conjectures.

If you know a counterexample, please let us know. We still plan to add the Szekeres-Wilf bound. Dan Cranston suggests this will remove the first conjecture below.

1. chromatic_number(x) <= maximum_average_degree(x) + 1

2. chromatic_number(x) <= clique_number(x) + girth(x)

3. chromatic_number(x) <= clique_number(x) + domination_number(x)

4. chromatic_number(x) <= 2*clique_number(x) + 1

5. chromatic_number(x) <= clique_number(x) + 1/2*max_degree(x)

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