Weak degeneracy of regular graphs

Motivated by the study of greedy algorithms for graph coloring, Bernshteyn and Lee introduced a generalization of graph degeneracy, which is called weak degeneracy. In this paper, we show the lower bound of the weak degeneracy for $d$-regular graphs is exactly $\lfloor d\mathrm{/}2\rfloor +1$, which is tight. This result refutes the conjecture of Bernshteyn and Lee on this lower bound.