On the independence and chromatic numbers of random regular graphs

AbstractLet Gr denote a random r-regular graph with vertex set {1, 2, …, n} and α(Gr) and χ(Gr) denote respectively its independence and chromatic numbers. We show that with probability going to 1 as n → ∞ respectively |δ(Gr) − 2nr(logr − log logr + 1 − log 2)|⩽γnr and |χ(Gr) − r2 log r − 8r log logr(log)2| ⩽ 8r log log r(log r)2 provided r = o(nθ), θ < 13, 0 < ε < 1, are constants, and r ≥ rε, where rε depends on ε only

H\"older-type inequalities and their applications to concentration and correlation bounds

Let $Y_v, v\in V,$ be $[0,1]$-valued random variables having a dependency graph $G=(V,E)$. We show that $$\mathbb{E}\left[\prod_{v\in V} Y_{v} \right] \leq \prod_{v\in V} \left\{ \mathbb{E}\left[Y_v^{\frac{\chi_b}{b}}\right] \right\}^{\frac{b}{\chi_b}},$$ where $\chi_b$ is the $b$-fold chromatic number of $G$. This inequality may be seen as a dependency-graph analogue of a generalised H\"older inequality, due to Helmut Finner. Additionally, we provide applications of H\"older-type inequalities to concentration and correlation bounds for sums of weakly dependent random variables.Comment: 15 page