Problems with Unknown Single Exponential Time Agorithms
In the Graph Homomorphism problem, the input is two graphs $G$ and $H$ and the question is whether there is a mapping $h$ from the vertices of $G$ to the vertices of $H$ such that for every edge $uv\in E(G)$ we have that $h(u) h(v)\in E(H)$.

The problem can be solved in time $O^*(|V(H)|^{|V(G)})$ by a brute-force algorithm (the $O^*$-notation hides factors polynomial in the input size).

However it is open whether it can be solved in time $O^*(c^{|V(H)|+|V(G)|})$, and this appears as an open question in


Fedor V. Fomin, Pinar Heggernes, Dieter Kratsch:
Exact Algorithms for Graph Homomorphisms. Theory Comput. Syst. 41(2): 381-393 (2007)