Reed's Conjecture on hole expansions

A note on Reed's Conjecture about $ω$, $Δ$ and $χ$ with respect to vertices of high degree

A note on Reed's Conjecture about $\omega$, $\Delta$ and $\chi$ with respect to vertices of high degree

Reed conjectured that for every graph, χ ≤ ∆+ω+1 2 holds, where χ, ω and ∆ denote the chromatic number, clique number and maximum degree of the graph, respectively. We develop an algorithm which takes a hypothetical counterexample as input. The output discloses some hidden structures closely related to high vertex degrees. Consequently, we deduce two graph classes where Reed's Conjecture holds:...

A Note On Reed's Conjecture

In [5], Reed conjectures that every graph satisfies χ ≤ ⌈ω+∆+1 2 ⌉ . We prove this holds for graphs with disconnected complement. Combining this fact with a result of Molloy proves the conjecture for graphs satisfying χ > ⌈ n 2 ⌉ . Generalizing this we prove that the conjecture holds for graphs satisfying χ > n+3−α 2 . It follows that the conjecture holds for graphs satisfying ∆ ≥ n + 2 − (α +√...

Reed's conjecture on some special classes of graphs

Reed conjectured that for any graph G, χ(G) ≤ ⌈ ω(G)+∆(G)+1 2 ⌉, where χ(G), ω(G), and ∆(G) respectively denote the chromatic number, the clique number and the maximum degree of G. In this paper, we verify this conjecture for some special classes of graphs, in particular for subclasses of P5-free graphs or Chair-free graphs.

A Superlocal Version of Reed's Conjecture

Reed’s well-known ω, ∆, χ conjecture proposes that every graph satisfies χ ≤ ⌈ 1 2 (∆ + 1 + ω)⌉. The second author formulated a local strengthening of this conjecture that considers a bound supplied by the neighbourhood of a single vertex. Following the idea that the chromatic number cannot be greatly affected by any particular stable set of vertices, we propose a further strengthening that con...