Some results on Reed’s Conjecture about [formula omitted], and [formula omitted] with respect to [formula omitted

Abstract

Let G be a graph of order n with chromatic number χ , maximum degree Δ , clique number ω and independence number α . In 1998 Reed conjectured that χ is bounded from above by ⌈ Δ + ω + 1 2 ⌉ . We will present some partial solutions for this conjecture with respect to α . For instance, we will verify Reed’s Conjecture for graphs with independence number α = 2 , for graphs with maximum degree Δ ≥ n − α − 4 , and for triangle-free graphs having maximum degree Δ ≥ 8 ( n − α ) + 118 21 . In addition, we will prove the general upper bound χ ≤ 1 3 ( n − α + ω + 2 + Δ + ω + 1 2 ) .