The $L(2,1)$-$\cal F$-Labeling Problem of Graphs

Abstract

In order to unify various concepts of distance-two labelings, we consider a general setting of distance-two labelings as follows. Given a graph $H$, an $L(2,1)$-$H$-labeling of a graph $G$ is a mapping $f$ from $V(G)$ to $V(H)$ such that $d_H(f(u),f(v)) \ge 2$ if $d_G(u,v) = 1$ and $d_H(f(u),f(v)) \ge 1$ if $d_G(u,v) = 2$. Suppose $\cal F$ is a family of graphs. The $L(2,1)$-${\cal F}$-labeling problem is to determine the $L(2,1)$-${\cal F}$-labeling number $\lambda_{\cal F}(G)$ of a graph $G$ which is the smallest number $|E(H)|$ such that $G$ has an $L(2,1)$-$H$-labeling for some $H \in {\cal F}$. Notice that the $L(2,1)$-${\cal F}$-labeling is the $L(2,1)$-labeling (respectively, the circular distance-two labeling) if ${\cal F}$ is the family of all paths (respectively, cycles). The purpose of this paper is to study the $L(2,1)$-${\cal F}$-labeling problem.