In this paper, we give a sufficient condition for graph choosability, based on Combinatorial Nullstellensatz and a specific property, called “double list colorability”, which means that there is a list assignment for which there are exactly two admissible colorings.

We show that triangle-free penny graphs have degeneracy at most two, list coloring number (choosability) at most three, diameter D = Ω( √ n), and at most min ( 2n−Ω( √ n), 2n−D − 2 ) edges.

We study circular choosability, a notion recently introduced by Mohar and by Zhu. First, we provide a negative answer to a question of Zhu about circular cliques. We next prove that cch(G) = O (ch(G) + ln |V (G)|) for every graph G. We investigate a generalisation of circular choosability, the circular f -choosability, where f is a function of the degrees. We also consider the circular choice n...

We study a restricted form of list colouring, for which every pair of lists that correspond to adjacent vertices may not share more than one colour. The optimal list size such that a proper list colouring is always possible given this restriction, we call separation choosability. We show for bipartite graphs that separation choosability increases with (the logarithm of) the minimum degree. This...

Equitable k-choosability is a list analogue of equitable k-coloring that was introduced by Kostochka et al. (J Graph Theory 44:166–177, 2003). It known if vertex disjoint graphs $$G_1$$ and $$G_2$$ are equitably k-choosable, the union may not be k-choosable. Given any $$m \in \mathbb {N}$$ values k for which $$K_{1,m}$$ k-choosable known. Also, complete characterization 2-choosable With these f...