Finding Large $H$-Colorable Subgraphs in Hereditary Graph Classes

We study the Max Partial $H$-Coloring problem: given a graph $G$, find largest induced subgraph of $G$ that admits homomorphism into $H$, where $H$ is fixed pattern without loops. Note when complete on $k$ vertices, problem reduces to finding $k$-colorable subgraph, which for $k=2$ equivalent (by complementation) Odd Cycle Transversal. prove every loops, can be solved in $\{P_5,F\}$-free graphs polynomial time, whenever $F$ threshold graph; $\{P_5,{bull}\}$-free time; $P_5$-free time $n^{\mathcal{O}(\omega(G))}$; and $\{P_6,{1-subdivided claw}\}$-free $n^{\mathcal{O}(\omega(G)^3)}$. Here, $n$ number vertices input $\omega(G)$ maximum size clique $G$. Furthermore, by combining mentioned algorithms with simple branching procedure, we obtain subexponential-time these classes graphs. Finally, show even restricted variant $\mathsf{NP}$-hard considered subclasses if allow loops $H$.