I am studying graphs $G$ for which $\chi(H)<\chi(G)$ for all minors $H$ of $G$. Complete graphs are examples of such graphs and bipartite graphs (except $K_2$) are nonexamples.
Are there any characterizations or nice properties of this kind of graphs? In particular we may look at the ones on $n$ vertices with minimum number of edges. I would appreciate your thoughts or relevant references.
Hadwiger's conjecture is equivalent to the claim that complete graphs are the only such graphs.
In one direction: if a graph $G$ with $\chi(G) = k$ has $K_k$ as a minor, then it has a minor $H$ with $\chi(H) = \chi(G)$, so it fails to have this property. So if Hadwigers'c conjecture holds, the only graphs with this property are complete graphs.
In the other: if complete graphs are the only graphs with this property, then for any other graph $G$ we may pass to a minor $H$ with $\chi(H) \ge \chi(G)$, and repeat until we are left with a complete graph, which implies Hadwiger's conjecture.
All such graphs, if they exist, must be critical graphs. This ensures, also, that contracting any one edge reduces the chromatic number. However, it may be that contracting further edges later on will bring the chromatic number back up.