A Note on the Gyárfás–Sumner Conjecture

Abstract

The Gyárfás–Sumner conjecture says that for every tree T and every integer \(t\ge 1\), if G is a graph with no clique of size t and with sufficiently large chromatic number, then G contains an induced subgraph isomorphic to T. This remains open, but we prove that under the same hypotheses, G contains a subgraph H isomorphic to T that is “path-induced”; that is, for some distinguished vertex r, every path of H with one end r is an induced path of G.