Sebő’s 3‑colorability conjecture for triangle‑free t‑perfect graphs

Establish whether every triangle‑free t‑perfect graph is 3‑colorable; that is, prove that for every graph G that is both triangle‑free and t‑perfect, there exists a proper vertex 3‑coloring of G.

Background

T‑perfect graphs are defined via the equality between their stable set polytope and the polytope given by nonnegativity, edge, and odd cycle inequalities. While general t‑perfect graphs are known not to be universally 3‑colorable (with explicit 4‑critical counterexamples), the triangle‑free subclass is conjectured to admit a 3‑coloring. This conjecture remains unresolved and is highlighted early in the paper as a longstanding open problem.