the question I'm concerned with is twofold.
First: I'm wondering about the relationship between complete subgraphs and $k$-degeneracy.
Let $G$ be an undirected simple graph.
A regular subgraph of magnitude $k+1$,does have a subgraph with vertices of degree $\delta(v) = k$ and therefore degeneracy of $k$. Does a graph have to have a regular subgraph of magnitude $k+1$ to have a degeneracy of $k$?
I have'nt found that fact online and therefore this seems very suspicious to me.
Secondly: I am supposed to prove that subdividing an edge in a graph( actually means splitting the edge and adding an inbetween vertex) does not change it's treewidth. And by virtue of that fact to conclude that $2$-degenerate graphs do have unbounded treewidth.
This seems truely strange to me, for e.g.:
Consider a triangle between vertices $u,v,w$. Now simply add a vertex $u'$ between $u$ and $v$, delete the edge $(u,v)$ and add two edges $(u,u')$ and $(u',v)$. The resulting graph is still $2$-degenrate and still has the same treewidth. But what i'm looking for actually is a constellation where i do increase the treewidth, without increasing the degeneracy?! right? I don't see any such opportunity.
Thanks a lot!