User blog comment:Wythagoras/Graphs/@comment-1605058-20140913123957/@comment-10429372-20140913135622

Yes, your're right.

However, I think Deedlit's method is also incorrect:


 * Also, I believe I can prove that SSCG(3n+4) >= 3 SCG(n), improving Goucher's inequality. We can convert any multigraph into a simple graph by the following procedure: Whenever we have multiple edges between two vertices, we subdivide all but one of the edges into two edges. Whenever we have a loop, we subdivide it into three edges. Each vertex either has a loop, a double or triple edge, or neither. In any case we need to add at most two vertices to the adjoining edges. So this conversion at most triples the number of edges.


 * So, let's say we have a sequence of SCG(n) multigraphs with at most n+1, n+2, ... n+SCG(n) vertices. The conversion converts this to a sequence of simple graphs with at most 3n+3, 3n+6, ... 3n + 3 SCG(n) vertices. Since each subsequent graph in the sequence has the max number of vertices increasing by 3, we replace each graph by a sequence of three graphs, by replacing G by G + 2 vertices, G + 1 vertex, G. This leads to a sequence with 3n+5, 3n+4, 3n+3, 3n+8, 3n+7, 3n+6,... vertices. This is a valid sequence for SSCG(3n+4) of length 3SCG(n), so SSCG(3n+4) >= 3SCG(n). Deedlit11 (talk) 16:12, June 28, 2013 (UTC)