Beyond Ohba's Conjecture: A bound on the choice number of $k$-chromatic graphs with $n$ vertices

Let $\text{ch}(G)$ denote the choice number of a graph $G$ (also called “list chromatic number” or “choosability” of $G$). Noel, Reed, and Wu proved the conjecture of Ohba that $\text{ch}(G)=\chi (G)$ when $\mathrm{\mid }V(G)\mathrm{\mid }\le 2\chi (G)+1$. We extend this to a general upper bound: $$\text{ch}$(G)\le \max{\chi(G),\lceil({|V(G)|+\chi(G)-1})/{3}\rceil}$\mathrm{.}Ourresultissharpfor$|V(G)|\le 3\chi(G)$ using Ohba’s examples, and it improves the best-known upper bound for $\text{ch}(K_{4,\dots ,4})$.