Coloring graphs of various maximum degree from random lists

Let $G=G(n)$ be a graph on $n$ vertices with maximum degree $Δ=Δ(n)$. Assign to each vertex $v$ of $G$ a list $L(v)$ of colors by choosing each list independently and uniformly at random from all $k$-subsets of a color set $\mathcal{C}$ of size $σ= σ(n)$. Such a list assignment is called a \emph{random $(k,\mathcal{C})$-list assignment}. In this paper, we are interested in determining the asymptotic probability (as $n \to \infty$) of the existence of a proper coloring $φ$ of $G$, such that $φ(v) \in L(v)$ for every vertex $v$ of $G$, a so-called $L$-coloring. We give various lower bounds on $σ$, in terms of $n$, $k$ and $Δ$, which ensures that with probability tending to 1 as $n \to \infty$ there is an $L$-coloring of $G$. In particular, we show, for all fixed $k$ and growing $n$, that if $σ(n) = ω(n^{1/k^2} Δ^{1/k})$ and $Δ=O\left(n^{\frac{k-1}{k(k^3+ 2k^2 - k +1)}}\right)$, then the probability that $G$ has an $L$-coloring tends to 1 as $n \rightarrow \infty$. If $k\geq 2$ and $Δ= Ω(n^{1/2})$, then the same conclusion holds provided that $σ=ω(Δ)$. We also give related results for other bounds on $Δ$, when $k$ is constant or a strictly increasing function of $n$.

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