In this paper we relate a fundamental parameter of a random graph, its degree sequence, to a simple model of nearly independent binomial random variables. This confirms a conjecture made in 1997. As a result, many interesting functions of the joint distribution of graph degrees, such as the distribution of the median degree, become amenable to estimation. Our result is established by proving an asymptotic formula conjectured in 1990 for the number of graphs with given degree sequence. In particular, this gives an asymptotic formula for the number of d-regular graphs for all d, as n Ñ 8.

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A graph G is r-Ramsey for a graph H, denoted by G → (H) r , if every r-colouring of the edges of G contains a monochromatic copy of H. The graph G is called r-Ramseyminimal for H if it is r-Ramsey for H but no proper subgraph of G possesses this property. Let s r (H) denote the smallest minimum degree of G over all graphs G that are r-Ramseyminimal for H. The study of the parameter s 2 was initiated by Burr, Erdős, and Lovász in 1976 when they showed that for the clique s 2 (K k ) = (k − 1) 2 . In this paper, we study the dependency of s r (K k ) on r and show that, under the condition that k is constant, s r (K k ) = r 2 · polylog r. We also give an upper bound on s r (K k ) which is polynomial in both r and k, and we determine s r (K 3 ) up to a factor of log r.

Abstract. In this paper we study the behaviour of the domination number of the Erdős-Rényi random graph G(n, p). Extending a result of Wieland and Godbole we show that the domination number of G(n, p) is equal to one of two values asymptotically almost surely whenever p ≫The explicit values are exactly at the first moment threshold, that is where the expected number of dominating sets starts to tend to infinity. For small p we also provide various non-concentration results which indicate why some sort of lower bound on the probability p is necessary in our first theorem. Concentration, though not on a constant length interval, is proven for every p ≫ 1/n. These results show that unlike in the case ofwhere concentration of the domination number happens around the first moment threshold, for p = O(ln n/n) it does so around the median. In particular, in this range the two are far apart from each other.