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

TL;DR

This paper presents a faster and simpler algorithm for coloring a specific subclass of graphs that contain a 7-cycle, leveraging neighborhood diversity instead of clique-width, improving efficiency over previous methods.

Contribution

The authors introduce a more efficient coloring algorithm for $(4K_1, C_4, C_6)$-free graphs with a $C_7$, based on neighborhood diversity, simplifying and speeding up prior approaches.

Findings

The new algorithm is significantly faster than previous methods.

It effectively uses neighborhood diversity as a key parameter.

The approach simplifies the coloring process for the graph class.

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.