Towards a Version of Ohba’s Conjecture for Improper Colorings

Abstract

The d-improper chromatic number $\chi ^d(G)$ of a graph G is the minimum number of colors to color G such that each color class induces a subgraph of maximum degree at most d. The d-improper choice number is the list-coloring version of this concept. A graph is called d-improperly chromatic-choosable if its d-improper choice number equals its d-improper chromatic number. As a generalization of a recently confirmed conjecture of Ohba that every graph G with $|V(G)| \le 2\chi (G)+1$ is chromatic-choosable, Yan et al. proposed an improper coloring-based version of Ohba’s conjecture: every graph G with $|V(G)|\le (d+2)\chi ^d(G)+(d+1)$ is d-improperly chromatic-choosable. In this paper, using graph theoretic and probabilistic methods we prove that the conjecture is true for $|V(G)| \le (d+\frac{3}{2})\chi ^d(G)+\frac{d}{2}$. We also construct a family of graphs G with $|V(G)|=(d+3)\chi ^d(G)+(d+3)$ which are not d-improperly chromatic-choosable.