We prove that the adaptable chromatic number of a graph is at least asymptotic to the square root of the chromatic number. This is best possible. Consider a graph G where each edge of G is assigned a colour from {1, ..., k} (this is not necessarily a proper edge colouring). A k-adapted colouring is an assignment of colours from {1, ..., k} to the vertices of G such that there is no edge with th...

Let G be any triangle-free graph with maximum degree ∆ ≤ 3. Staton proved that the independence number of G is at least 5 14 n. Heckman and Thomas conjectured that Staton’s result can be strengthened into a bound on the fractional chromatic number of G, namely χf (G) ≤ 14 5 . Recently, Hatami and Zhu proved χf (G) ≤ 3− 3 64 . In this paper, we prove χf (G) ≤ 3− 3 43 .

We consider a number of related results taken from two papers – one by W. Lin [1], and the other D. C. Fisher[2]. These articles treat various forms of graph colorings and the famous Mycielski construction. Recall that the Mycielskian μ(G) of a simple graph G is a graph whose chromatic number satisfies χ(μ(G)) = χ(G)+1, but whose largest clique is no larger than the largest clique in G. Extendi...

We prove the conjecture of A. Postnikov that (A) the number of regions in the inversion hyperplane arrangement associated with a permutation w ∈ Sn is at most the number of elements below w in the Bruhat order, and (B) that equality holds if and only if w avoids the patterns 4231, 35142, 42513 and 351624. Furthermore, assertion (A) is extended to all finite reflection groups. A byproduct of thi...

A graph is h-perfect if its stable set polytope can be completely described by nonnegativity, clique and odd-hole constraints. It is t-perfect if it furthermore has no clique of size 4. For every graph G and every c ∈ Z (G) + , the weighted chromatic number of (G, c) is the minimum cardinality of a multi-set F of stable sets of G such that every v ∈ V (G) belongs to at least cv members of F . W...