I haven't worked through the details yet, but what if is the union of and a sufficiently huge bipartite graph ? Then , and by taking huge enough, you can get as close to 2 as you like, forcing as small as you like.
EDIT: Whoo boy. Nevermind.
No?
I haven't worked through the details yet, but what if is the union of and a sufficiently huge bipartite graph ? Then , and by taking huge enough, you can get as close to 2 as you like, forcing as small as you like.
EDIT: Whoo boy. Nevermind.