What are some important conjectures in graph theory that have been checked by computer up to order 11?
closed as not a real question by Bruce Westbury, Will Jagy, Ryan Budney, Chandan Singh Dalawat, Gerry Myerson Jan 31 '12 at 11:45
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Graffiti (by S. Fajtlowicz) and Graffiti.pc (by E. DeLaViña) are computer programs that produce conjectures in graph theory.
See Doug West's web page, "Some Conjectures of Graffiti.pc (2004-07)," and the more recent, "Bibliography on Conjectures, Methods and Applications of Graffiti," which includes papers through 2011.
I know one. The reconstruction conjecture of the undirected graph. That work is due to BD McKay.