If $\ X\ $ is a set, we let $\ \binom X2\,=\, \big\{\{a,b\}: a \neq b \in X \big\}.\ $ Given a simple, undirected graph $\ G=(V,E),\ $ we let $\ \delta(G)\ $ be its minimum degree, and $\ \Delta(G)\ $ its maximum degree. We say that $\ G\ $ is *self-complementary* if $\ G \cong \bar{G}\ $ where $\ \bar{G} = \left(V, \binom V2\setminus E\right)$. 


Given $\ N\in\mathbb{N},\ $ is there a self-complementary graph $\ G\ $ with $\ \Delta(G) \geq N\cdot \delta(G)\,$?



Minimal and maximal degrees in self-complementary graph
Self-complementary graphs exist if $n\equiv0,1\pmod 4$. 


Take a self-complementary graph $G$ with $n$ vertices. Append to it a path $u{-}v{-}w{-}x$ of 4 vertices and join $v$ and $w$ to all of $G$. Now you have a self-complementary graph with $n+4$ vertices, $\delta=1$ and $\Delta=n+2$. This gives all sizes from 4 onwards.


