I have an idea about simple graphs that seems to check out for all cases which I have randomly generated on my computer. They were generated over a variety of vertex counts, where I included edges with some probability $p$, which I tested for various values.
I know that up to 10 vertices, there are not too many graphs (see oeis). Also, there are many specific graph which have served as counterexamples to other peoples ideas in the past.
Are there any good lists of graphs that are freely available online? I feel like checking many randomly generated graphs might be repeating lots of work.