321 research outputs found

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

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    Let k:=(k1,,ks)\mathbf{k} := (k_1,\dots,k_s) be a sequence of natural numbers. For a graph GG, let F(G;k)F(G;\mathbf{k}) denote the number of colourings of the edges of GG with colours 1,,s1,\dots,s such that, for every c{1,,s}c \in \{1,\dots,s\}, the edges of colour cc contain no clique of order kck_c. Write F(n;k)F(n;\mathbf{k}) to denote the maximum of F(G;k)F(G;\mathbf{k}) over all graphs GG on nn 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 k\mathbf{k} and nn, there is a complete multipartite graph GG on nn vertices with F(G;k)=F(n;k)F(G;\mathbf{k}) = F(n;\mathbf{k}). Also, for every k\mathbf{k} we construct a finite optimisation problem whose maximum is equal to the limit of log2F(n;k)/(n2)\log_2 F(n;\mathbf{k})/{n\choose 2} as nn tends to infinity. Our final result is a stability theorem for complete multipartite graphs GG, describing the asymptotic structure of such GG with F(G;k)=F(n;k)2o(n2)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

    Computing the chromatic number of t-(v,k,[lambda]) designs

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    Colouring t-designs has previously been shown to be an NP-complete problem; heuristics and a practical algorithm for this problem were developed for this thesis; the algorithm was then employed to find the chromatic numbers of the sixteen non- isomorphic 2-(25, 4, 1) designs and the four cyclic 2-(19, 3, 1) designs. This thesis additionally examines the existing literature on colouring and finding chromatic numbers of t-designs

    New bounds for odd colourings of graphs

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    Given a graph GG, a vertex-colouring σ\sigma of GG, and a subset XV(G)X\subseteq V(G), a colour xσ(X)x \in \sigma(X) is said to be \emph{odd} for XX in σ\sigma if it has an odd number of occurrences in XX. We say that σ\sigma is an \emph{odd colouring} of GG if it is proper and every (open) neighbourhood has an odd colour in σ\sigma. The odd chromatic number of a graph GG, denoted by χo(G)\chi_o(G), is the minimum kNk\in\mathbb{N} such that an odd colouring σ ⁣:V(G)[k]\sigma \colon V(G)\to [k] exists. In a recent paper, Caro, Petru\v sevski and \v Skrekovski conjectured that every connected graph of maximum degree Δ3\Delta\ge 3 has odd-chromatic number at most Δ+1\Delta+1. We prove that this conjecture holds asymptotically: for every connected graph GG with maximum degree Δ\Delta, χo(G)Δ+O(lnΔ)\chi_o(G)\le\Delta+O(\ln\Delta) as Δ\Delta \to \infty. We also prove that χo(G)3Δ/2+2\chi_o(G)\le\lfloor3\Delta/2\rfloor+2 for every Δ\Delta. If moreover the minimum degree δ\delta of GG is sufficiently large, we have χo(G)χ(G)+O(ΔlnΔ/δ)\chi_o(G) \le \chi(G) + O(\Delta \ln \Delta/\delta) and χo(G)=O(χ(G)lnΔ)\chi_o(G) = O(\chi(G)\ln \Delta). Finally, given an integer h1h\ge 1, we study the generalisation of these results to hh-odd colourings, where every vertex vv must have at least min{deg(v),h}\min \{\deg(v),h\} odd colours in its neighbourhood. Many of our results are tight up to some multiplicative constant

    Inapproximability of counting hypergraph colourings

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