The uniquely list chromatic number bound and equality characterization

For a graph $G$ and a positive integer $k$, let $\chi_u(G,k)$ be the minimum number $t$ such that $G$ is uniquely $(k,t)$-list colorable, and let $\chi_u(G)=\max_{k\geqslant 1}\chi_u(G,k)$. Let $\Delta(G)$ denote the maximum degree of $G$. The uniquely list chromatic number conjecture. For every graph $G$,

$$\chi_u(G)\leqslant\Delta(G)+1,$$

and equality holds if and only if $G$ is either a complete graph or an odd cycle. This extends the result established in the paper for uniquely $2$-list colorable graphs; the conjecture proposes the corresponding bound for all positive integers $k$, together with a characterization of the equality cases.