Polynomial-time solvability of Max Partial H-Coloring on (bull, E)-free graphs

Let $H$ be a simple graph. In \Max Partial $H$-Coloring\, the input is a graph $G$ with a revenue function $\mathsf{rev}:V(G)\times V(H)\to\mathbb{Q}_{\geq 0}$, and the task is to choose a set $X\subseteq V(G)$ and an $H$-coloring $c$ of $G[X]$ maximizing

$$\sum_{v\in X}\mathsf{rev}(v,c(v)).$$

Here, a graph is $(\textrm{bull},\textsf{E})$-free if it has no induced subgraph isomorphic to the bull or to $\textsf{E}$. Strengthening of the preceding results. For every simple graph $H$, \Max Partial $H$-Coloring\ on $(\textrm{bull},\textsf{E})$-free graphs can be solved in polynomial time. The preceding algorithmic results establish a related bound for every simple graph $H$ on $(\textrm{bull},\textsf{E})$-free instances with $n$ vertices and clique number $\omega$, but the claimed polynomial-time bound for arbitrary simple $H$ is presented as a stronger conjectural variant; graphs $H$ with loops are excluded because corresponding problems are already NP-hard on complements of bipartite graphs.