Some new perspectives on distance two labeling

An $L(2,1)$-labeling (or distance two labeling) of a graph $G$ is a function $f$ from the vertex set $V(G)$ to the set of nonnegative integers such that $|f(u)-f(v)|geq2$ if $d(u,v)=1$ and $|f(u)-f(v)|geq1$ if $d(u,v)=2$. The $L(2,1)$-labeling number $lambda(G)$ of $G$ is the smallest number $k$ such that $G$ has an $L(2,1)$-labeling with max${f(v):v in V(G)}=k$. In this paper we find $lambda$-number for some cacti.