Graphs with vanishing Betti numbers β_{i,j}(I_G)=0 for fixed i,j have chromatic number at most a polynomial of degree 2j-2i-4 in the clique number ω.
Algebraic Properties of Edge Ideals via Combinatorial Topology.The Electronic Journal of Com- binatorics16 (2009)
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Chromatic numbers from edge ideals: Graph classes with vanishing syzygies are polynomially $\chi$-bounded
Graphs with vanishing Betti numbers β_{i,j}(I_G)=0 for fixed i,j have chromatic number at most a polynomial of degree 2j-2i-4 in the clique number ω.