Abstract
In this article we show that Maximum Partial List H-Coloring is polynomial-time solvable on P_5-free graphs for every fixed graph H. In particular, this implies that Maximum k-Colorable Subgraph is polynomial-time solvable on P_5-free graphs. This answers an open question from Agrawal, Lima, Lokshtanov, Saurabh & Sharma [SODA 2024]. This also improves the $n^{O(\omega(G))}$-time algorithm for Maximum Partial H-Coloring by Chudnovsky, King, Pilipczuk, Rz\k{a}\.{z}ewski & Spirkl [SIDMA 2021] to polynomial-time algorithm.