Some remarks on graphs with no induced subdivision of [formula omitted

Abstract

For a positive integer n, we use Kn and Pn to denote a complete graph and an induced path on n vertices, respectively. A subdivision of G is a graph obtained from G by replacing the edges of G with independent paths of length at least one between their end vertices. A graph is said to be ISK4-free if it does not contain any subdivision of K4 as an induced subgraph. Lévêque, Maffray and Trotignon conjectured that every ISK4-free graph is 4-colorable. In this paper, we show that this conjecture is true for the class of P6-free graphs.