On two problems of defective choosability of graphs

Abstract

Given positive integers $p\ge k$, and a nonnegative integer $d$, we say a graph $G$ is $\left(k,d,p\right)$-choosable if for every list assignment $L$ with $\mid L\left(v\right)\mid \ge k$ for each $v\in V\left(G\right)$ and $\mid {\bigcup }_{v\in V\left(G\right)}L\left(v\right)\mid \le p$, there exists an $L$-coloring of $G$ such that each monochromatic subgraph has maximum degree at most $d$. In particular, $\left(k,0,k\right)$-choosable means $k$-colorable, $\left(k,0,+\infty \right)$-choosable means $k$-choosable and $\left(k,d,+\infty \right)$-choosable means $d$-defective $k$-choosable. This paper proves that there are 1-defective 3-choosable planar graphs that are not 4-choosable, and for any positive integers $\ell \ge k\ge 3$, and nonnegative integer $d$, there are $\left(k,d,\ell \right)$-choosable graphs that are not $\left(k,d,\ell +1\right)$-choosable. These results answer questions asked by Wang and Xu, and Kang, respectively. Our construction of $\left(k,d,\ell \right)$-choosable but not $\left(k,d,\ell +1\right)$-choosable graphs generalizes the construction of Král' and Sgall for the case $d=0$.