A
k-L(2,1)-labeling for a graph
G is a function

such that

whenever

and

whenever
u and
v are at distance two apart. The
λ-number for
G, denoted by
λ(
G), is the minimum
k over all
k-L(2,1)-labelings of
G. In this paper, we show that

for

or 11, which confirms Conjecture 6.1 stated in [X. Li, V. Mak-Hau, S. Zhou, The
L(2,1)-labelling problem for cubic Cayley graphs on dihedral groups, J. Comb. Optim. (2013) 25: 716-736] in the case when

or 11. Moreover, we show that

if 1) either

(mod 6),
m is odd,
r = 3, or 2)

(mod 3),
m is even (mod 2),
r = 0.