# The $L(2,1)$-choosability‎ ‎of cycle

Document Type: Research Paper

Authors

1 Hong Kong Baptist University

2 United International College

Abstract

‎For a given graph $G=(V,E)$‎, ‎let $\mathscr L(G)=\{L(v)‎ : ‎v\in V\}$ be a prescribed list assignment‎. ‎$G$ is $\mathscr L$-$L(2,1)$-colorable if there exists a vertex labeling $f$ of $G$ such that $f(v)\in L(v)$ for all $v \in V$; $|f(u)-f(v)|\geq 2$ if $d_G(u,v) = 1$; and $|f(u)-f(v)| \geq 1$ if $d_G(u,v)=2$‎. ‎If $G$ is $\mathscr L$-$L(2,1)$-colorable for every list assignment $\mathscr L$ with $|L(v)|\geq k$ for all $v\in V$‎, ‎then $G$ is said to be $k$-$L(2,1)$-choosable‎. ‎In this paper‎, ‎we prove all cycles are $5$-$L(2,1)$-choosable‎.

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