Towards a Version of Ohba’s Conjecture for Improper Colorings

Abstract

The d-improper chromatic number $$\chi ^d(G)$$?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 $$|V(G)| \le 2\chi (G)+1$$|V(G)|≤2?(G)+1 is chromatic-choosable, Yan et al. proposed an improper coloring-based version of Ohba's conjecture: every graph G with $$|V(G)|\le (d+2)\chi ^d(G)+(d+1)$$|V(G)|≤(d+2)?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 $$|V(G)| \le (d+\frac{3}{2})\chi ^d(G)+\frac{d}{2}$$|V(G)|≤(d+32)?d(G)+d2. We also construct a family of graphs G with $$|V(G)|=(d+3)\chi ^d(G)+(d+3)$$|V(G)|=(d+3)?d(G)+(d+3) which are not d-improperly chromatic-choosable.