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 \(\lambda_l(G) \leq \Delta(G) + 6\) if \(G\) is an outerplanar graph or \(A\)-graph; and \(\lambda_l(G) \leq \Delta(G) + 9\) if \(G\) is an \(HA\)-graph or Halin graph.