On Improperly Chromatic-Choosable Graphs

Abstract

A graph is $d$-improperly chromatic-choosable if its $d$-improper choice number coincides with its $d$-improper chromatic number. For fixed $d\ge 0$, we show that if the $d$-improper chromatic number is close enough to $\frac{1}{d+1}$ of the number of vertices in $G$, then $G$ is $d$-improperly chromatic-choosable. As a consequence, we show that the join $G + K_n$ is $d$-improperly chromatic-choosable when $n\ge (|V(G)|+d)^2$. We also raise a conjecture on $d$-improper chromatic-choosability.