Finding large $k$-colorable induced subgraphs in (bull, chair)-free and (bull,E)-free graphs

We study the Max Partial $k$-Coloring problem, where we are given a vertex-weighted graph, and we ask for a maximum-weight induced subgraph that admits a proper $k$-coloring. For $k=1$ this problem coincides with Maximum Weight Independent Set, and for $k=2$ the problem is equivalent (by complementation) to Minimum Odd Cycle Transversal. Furthermore, it generalizes $k$-Coloring. We show that Max Partial $k$-Coloring on $n$-vertex instances with clique number $\omega$ can be solved in time * $n^{\mathcal{O}(k\omega)}$ if the input graph excludes the bull and the chair as an induced subgraph, * $n^{\mathcal{O}(k\omega \log n)}$ if the input graph excludes the bull and E as an induced subgraph. This implies that $k$-Coloring can be solved in polynomial time in the former class, and in quasipolynomial-time in the latter one.