Suppose that graphs $A$ and $B$ with $V(A)=V(B)$ have Hadwiger numbers $a$ and $b$. That is, $K\_a$ and $K\_b$ are the largest clique minors of $A$ and $B$, respectively.
Are there upper bounds on the Hadwiger number of the union graph $A\cup B$ (with edge set $E(A\cup B)=E(A)\cup E(B)$) using $a$ and $b$ like $a+b$ or $a\cdot b$?



Bounding the size of clique minor of the union of two graphs
Take a complete graph $G\_n$, replace each edge $uv$ to a path $uxv$. Imagine that $ux\in A$, $vx\in B$, and that's for every edge. It looks that both $A,B$ are forests, so have small Hadwiger number.


