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Erd˝ os asked in 1962 about the value of f (n, k, l), the minimum number of k-cliques in a graph with order n and independence number less than l. The case (k, l) = (3, 3) was solved by Lorden. Here we solve the problem (for all large n) for (3, l) with 4 ≤ l ≤ 7 and (k, 3) with 4 ≤ k ≤ 7.

The k-domination number of a graph is the cardinality of a smallest set of vertices such that every vertex not in the set is adjacent to at least k vertices of the set. We prove two bounds on the k-domination number of a graph, inspired by two conjectures of the computer program Graffiti.pc. In particular, we show that for any graph with minimum degree at… (More)

Guessing games for directed graphs were introduced by Riis [8] for studying multiple unicast network coding problems. It can be shown that protocols for a multiple unicast network can be directly converted into a strategy for a guessing game. The performance of the optimal strategy for a graph is measured by the guessing number, and this number can be… (More)

Given a family of 3-graphs F, we define its codegree threshold coex(n, F) to be the largest number d = d(n) such that there exists an n-vertex 3-graph in which every pair of vertices is contained in at least d 3-edges but which contains no member of F as a subgraph. Let F 3,2 be the 3-graph on {a, b, c, d, e} with 3-edges abc, abd, abe and cde. In this… (More)

Guessing games for directed graphs were introduced by Riis [9] for studying multiple unicast network coding problems. In a guessing game, the players toss generalised dice and can see some of the other outcomes depending on the structure of an underlying digraph. They later guess simultaneously the outcome of their own die. Their objective is to find a… (More)

- E. R. Vaughan
- 2007

We will look at designs for microarray experiments, which have been studied by R. A. Bailey. These are designs of block size 2, that are really just graphs. We will look at the designs (graphs) that are said to be A-optimal, and investigate some intriguing links between A-optimality and other areas of mathematics. 1. Microarray experiments Microarray… (More)

For many positive odd integers n, whether prime or not, the set U n of units of Z n contains members t, u, v and w, say with respective orders τ , ψ, ω and π, such that we can write U n as the direct product U n = t × ×u × ×v × w. Each element of U n can then be written in the form t h u i v j w k where 0 ≤ h < τ, 0 ≤ i < ψ, 0 ≤ j < ω and 0 ≤ k < π. We can… (More)

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