Ordered colourings

On the Complexity of Ordered Colourings

Distance-regular graphs, MH-colourings and MLD-colourings

It is shown that the only bipartite distance regular graphs of diameter 3 which are MLDcolourable are the incidence graphs of complementary Hadamard designs.

Reconstruction of Random Colourings

Reconstruction problems have been studied in a number of contexts including biology, information theory and and statistical physics. We consider the reconstruction problem for random k-colourings on the ∆-ary tree for large k. Bhatnagar et. al. [1] showed non-reconstruction when ∆ ≤ 1 2 k log k − o(k log k) and reconstruction when ∆ ≥ k log k + o(k log k). We tighten this result and show nonrec...

Diagrams Colourings and Applications

This paper reviews some results involving arc colourings in a braid or a link diagram. The left self-distributivity idendity then appears as an algebraic counterpart to Reidemeister move of type III. Using classical and less classical self-distributive operations leads to a number of different results, as well as to many open questions. It has been observed for many years that there exist a con...

Planting Colourings Silently

Let k ≥ 3 be a fixed integer and let Zk(G) be the number of k-colourings of the graph G. For certain values of the average degree, the random variable Zk(G(n,m)) is known to be concentrated in the sense that 1 n (lnZk(G(n,m)) − lnE[Zk(G(n,m))]) converges to 0 in probability [Achlioptas and Coja-Oghlan: Proc. FOCS 2008]. In the present paper we prove a significantly stronger concentration result...