We study the \textsc{Max Partial H-Coloring} problem: given a graph G,
find the largest induced subgraph of G that admits a homomorphism into H,
where H is a fixed pattern graph without loops. Note that when H is a
complete graph on k vertices, the problem reduces to finding the largest
induced k-colorable subgraph, which for k=2 is equivalent (by
complementation) to \textsc{Odd Cycle Transversal}.
We prove that for every fixed pattern graph H without loops, \textsc{Max
Partial H-Coloring} can be solved:
∙ in {P5,F}-free graphs in polynomial time, whenever F is a
threshold graph;
∙ in {P5,bull}-free graphs in polynomial time;
∙ in P5-free graphs in time nO(ω(G));
∙ in {P6,1-subdivided claw}-free graphs in time
nO(ω(G)3).
Here, n is the number of vertices of the input graph G and ω(G) is
the maximum size of a clique in~G. Furthermore, combining the mentioned
algorithms for P5-free and for {P6,1-subdivided claw}-free
graphs with a simple branching procedure, we obtain subexponential-time
algorithms for \textsc{Max Partial H-Coloring} in these classes of graphs.
Finally, we show that even a restricted variant of \textsc{Max Partial
H-Coloring} is NP-hard in the considered subclasses of P5-free
graphs, if we allow loops on H