The Erd\H{o}s-Rothschild problem on edge-colourings with forbidden monochromatic cliques

Let $\mathbf{k} := (k_1,\dots,k_s)$ be a sequence of natural numbers. For a graph $G$, let $F(G;\mathbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,\dots,s$ such that, for every $c \in \{1,\dots,s\}$, the edges of colour $c$ contain no clique of order $k_c$. Write $F(n;\mathbf{k})$ to denote the maximum of $F(G;\mathbf{k})$ over all graphs $G$ on $n$ vertices. This problem was first considered by Erd\H{o}s and Rothschild in 1974, but it has been solved only for a very small number of non-trivial cases. We prove that, for every $\mathbf{k}$ and $n$, there is a complete multipartite graph $G$ on $n$ vertices with $F(G;\mathbf{k}) = F(n;\mathbf{k})$. Also, for every $\mathbf{k}$ we construct a finite optimisation problem whose maximum is equal to the limit of $\log_2 F(n;\mathbf{k})/{n\choose 2}$ as $n$ tends to infinity. Our final result is a stability theorem for complete multipartite graphs $G$, describing the asymptotic structure of such $G$ with $F(G;\mathbf{k}) = F(n;\mathbf{k}) \cdot 2^{o(n^2)}$ in terms of solutions to the optimisation problem.Comment: 16 pages, to appear in Math. Proc. Cambridge Phil. So

Fractional total colourings of graphs of high girth

Reed conjectured that for every epsilon>0 and Delta there exists g such that the fractional total chromatic number of a graph with maximum degree Delta and girth at least g is at most Delta+1+epsilon. We prove the conjecture for Delta=3 and for even Delta>=4 in the following stronger form: For each of these values of Delta, there exists g such that the fractional total chromatic number of any graph with maximum degree Delta and girth at least g is equal to Delta+1

Some Simple Distributed Algorithms for Sparse Networks

We give simple, deterministic, distributed algorithms for computing maximal matchings, maximal independent sets and colourings. We show that edge colourings with at most 2D-1 colours, and maximal matchings can be computed within O(log^* n + D) deterministic rounds, where D is the maximum degree of the network. We also show how to find maximal independent sets and (D+1)-vertex colourings within O(log^* n + D^2) deterministic rounds. All hidden constants are very small and the algorithms are very simple