Optimal frequency assignment and planar list L(2, 1)-labeling
2021
G has a list k-L(2, 1)-labeling if for any k-list assignment L, there exists a coloring $$c:V(G)\rightarrow \bigcup \limits _{v\in V} L(v)$$
of G such that $$c(v)\in L(v)$$
for $$\forall v\in V(G)$$
and for $$\forall u,v\in V(G)$$
, $$|c(u)-c(v)|\ge 2$$
if $$d(u,v)=1$$
, $$|c(u)-c(v)|\ge 1$$
if $$d(u,v)=2$$
. $$\lambda _{2,1}^{l}(G)=\min \{k|G$$
has a list k-L(2, 1)-labeling
$$\}$$
is called the list L(2, 1)-labeling number of G. In this paper, we prove that for planar graph with maximum degree $$\Delta \ge 5$$
, girth $$g\ge 13$$
and without adjacent 13-cycles, $$\lambda _{2,1}^{l}(G)\le \Delta +3$$
holds. Moreover, the upper bound $$\Delta +3$$
is tight.
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