Group coloring and group connectivity with non-isomorphic groups of the same order

Abstract

For every natural number $k$, there exists a planar graph which is $Z_2^k$-colorable, but not $\Gamma$-colorable for any other Abelian group $\Gamma$ of order $2^k$. Its dual graph is $Z_2^k$-connected, but not $\Gamma$-connected for any other Abelian group $\Gamma$ of order $2^k$.