Although the 2-choosable graphs have been fully classified in [2]

the 3-choosable graphs have not been classified yet

even if 3-choosable bipartite graphs. There are a few of results related it (see references [3] and [4]) and it seems difficult to completely classify the 3-choosable graphs. In this paper

we consider a weak problem

that is

to classify the [2

3]-choosable complete bipartite graphs. We prove that K3

7

K8

2 K1

n

K2

n(n≥1) and Km

n (m+n≤9 and m≠5 when n=4) are all [2

3]-choosable complete bipartite graphs

and thus the [2

3]-choosable complete bipartite graphs are completely characterized here

which

we consider

will be useful in the further research of the 3-choosable bipartite graphs.