The L(۲,۱)-choosability‎ ‎of cycle

‎For a given graph G=(V,E)‎, ‎let \mathscr L(G)=\{L(v)‎ : ‎v\in V\} be a prescribed list assignment‎. ‎G is \mathscr L-L(۲,۱)-colorable if there exists a vertex labeling f of G such that f(v)\in L(v) for all v \in V; |f(u)-f(v)|\geq ۲ if d_G(u,v) = ۱; and |f(u)-f(v)| \geq ۱ if d_G(u,v)=۲‎. ‎If G is \mathscr L-L(۲,۱)-colorable for every list assignment \mathscr L with |L(v)|\geq k for all v\in V‎, ‎then G is said to be k-L(۲,۱)-choosable‎. ‎In this paper‎, ‎we prove all cycles are ۵-L(۲,۱)-choosable‎.

Keywords:

L(۲ , ۱)-labeling , Choosability , cycle , Path