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.