Chromatic-number bounds for t-perfect graphs
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A graph is t-perfect when its stable-set polytope is described by nonnegativity, edge and odd-cycle inequalities. How many colors are always enough for this class? Shepherd’s conjectured bound is four.
Why it matters and what counts as progress
Why it matters: A universal bound links an exact polyhedral description of independent sets to graph coloring.
What counts as progress: A smaller proved universal color bound, such as an improvement from 186 to 50, or a proof or counterexample for the stronger four-color conjecture.
How the problem developed
- 1975 · Definition [1]
Václav Chvátal Defined t-perfect graphs through stable-set polytopes described by non-negativity, edge and odd-circuit inequalities.
- 1995 · Open question [1]
John Shepherd Asked whether t-perfect graphs have a finite chromatic-number bound; whether four colours always suffice remained open.
- 23 Dec 2024 · Bound [1]
Maria Chudnovsky, Linda Cook, James Davies, Sang-il Oum and Jane Tan Proved the first finite bound: every t-perfect graph is 199,053-colourable.
- 22 Oct 2026 · Improved bound [2]
Sang-il Oum Reduced the universal bound to 50 colours; the 4-colourability question remains open.
AI contribution
Every t-perfect graph is 50-colorable
Every finite t-perfect graph is 50-colorable. T-perfection means that its stable-set polytope is described by nonnegativity, edge and odd-cycle inequalities.
Sources
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