The paper [On graph classes with logarithmic boolean-width](https://arxiv.org/abs/1009.0216) claims that the boolean width of co-k-degenerate graphs is at most $k\log{n}$ and a lot of graph vertex partition problems can be solved in polynomial time. co-k-degenerate graphs include complements of bounded degree graphs. Clique is NP-hard on co-maximum degree 4. On the other hand, [graphclasses.org](http://graphclasses.org/classes/par_21.html) claims that clique is boolean width fixed parameter tractable, giving clique width as reference. Since $\exp{\log{n}}=n$ it could be polynomial. Are there complexity consequence of logarithmic boolean width of co-bounded degree graphs? Like ETH not holding for them?