A Note on Coloring $$(4K_1, C_4, C_6)$$-Free Graphs with a $$C_7$$

Abstract

Even-hole-free graphs are a graph class of much interest. Foley et al. (Graphs Comb. 36(1):125–138, 2020) have recently studied \((4K_1, C_4, C_6)\)-free graphs, which form a subclass of even-hole-free graphs. Specifically, Foley et al. have shown an algorithm for coloring these graphs via bounded clique-width if they contain a \(C_7\). In this note, we give a simpler and much faster algorithm via a more restrictive graph parameter, neighborhood diversity.