Brooks’ theorem states that for a graph G, if Δ (G)≥ 3 and ω (G)≤ Δ (G), then χ (G)≤ Δ (G). In this paper, we show that this chromatic bound can be further strengthened on (P 6, C 4, H)-free graphs, where H∈{b u l l, d i a m o n d}. In particular, we prove that for a (P 6, C 4, b u l l)-free graph G, if Δ (G)≥ 9 and ω (G)≤ Δ (G)− 1, then χ (G)≤ Δ (G)− 1. We also prove that for a (P 6, C 4, d i a m o n d)-free graph G, if Δ (G)≥ 5 and ω (G)≤ Δ (G)− 1, then χ (G)≤ Δ (G)− 1. We also show that similar results hold for (P 10, p a w)-free graphs and (P 5, C 5)-free graphs. These results validate the Borodin–Kostochka conjecture on these graph classes.