On List $(2,1)$-Labeling of Some Planar Graphs

Abstract

A list $(2,1)$-labeling $\mathcal{L}$ of graph $G$ is an assignment list $L(v)$ to each vertex $v$ of $G$ such that $G$ has a $(2,1)$-labeling $f$ satisfying $f(v)\in L(v)$ for all $v$ of graph $G$. If $|L(v)|=k+1$ for all $v$ of $G$, we say that $G$ has a $k$-list $(2,1)$-labeling. The minimum $k$ taken over all $k$-list $(2,1)$-labelings of $G$, denoted ${\lambda }_l(G)$, is called the list label-number of $G$. In this paper, we study the upper bound of $\lambda (G)$ of some planar graphs. It is proved that ${\lambda }_l(G)\le \mathrm{\Delta }(G)+6$ if $G$ is an outerplanar graph or $A$-graph; and ${\lambda }_l(G)\le \mathrm{\Delta }(G)+9$ if $G$ is an $HA$-graph or Halin graph.