Some results on Reed’s Conjecture about ω,Δω,Δ, and χχ with respect to αα

Let GG be a graph of order nn with chromatic number χχ, maximum degree ΔΔ, clique number ωω and independence number αα. In 1998 Reed conjectured that χχ is bounded from above by ⌈Δ+ω+12⌉. We will present some partial solutions for this conjecture with respect to αα. For instance, we will verify Reed’s Conjecture for graphs with independence number α=2α=2, for graphs with maximum degree Δ≥n−α−4Δ≥n−α−4, and for triangle-free graphs having maximum degree Δ≥8(n−α)+11821. In addition, we will prove the general upper bound χ≤13(n−α+ω+2+Δ+ω+12).