Two-colouring all two-element maximal antichains

Complexity of clique‐coloring odd‐hole‐free graphs

“…Indeed, many subclasses of odd-hole-free graphs have been studied and turn out to have a bounded clique-chromatic number: comparability graphs were proved to be all 2-clique-colorable [11] and cocomparability graphs are all 3-clique-colorable [10]. Strongly perfect graphs are obviously 2-clique-colorable since by definition they have a stable set that intersects all maximal cliques.…”

“…Among all the open questions about clique-coloring, the following one has arisen concerning perfect graphs: Question 1 (Duffus et al [11], Jensen and Toft [16,Problem 15.15]) Does there exist k 0 such that all perfect graphs are k 0 -clique-colorable? FIGURE 1.…”

When Will Every Maximal F-free Subposet Contain a Maximal Element?

“…For example, when F = 2, an F-free subposet of a poset P is just an antichain in P, and so ac 2 (P) is the minimum number of colours needed to colour the elements of P such that every maximal antichain with at least two elements receives at least two colours. It is proved in [2] that ac 2 (P) ≤ 3 for every poset P. (Moreover, three colours are sometimes necessary; an example is given in [3].) Similarly, if F = 2, then ac 2 (P) is the minimum number of colours needed to colour the elements of P such that each maximal chain of P with at least two elements receives at least two colours.…”

Packing and covering k-chain free subsets in Boolean lattices

“…[6,5,4,3,9,8,11]. Duffus et al [3] proved that τ 2 (P) ≤ 2 3 |P|, while in [11] it was shown that the coefficient 2 3 cannot be improved below 8 15 .…”