The $L(2,1)$-Labeling Problem on Graphs

• Authors: Gerard J. Chang;David Kuo
• Venue: SIAM Journal on Discrete Mathematics
• Year: 1996

An $L(2,1)$-labeling of a graph $G$ is a function $f$ from the vertex set $V(G)$ to the set of all nonnegative integers such that $|f(x)-f(y)|\ge 2$ if $d(x,y)=1$ and $|f(x)-f(y)|\ge 1$ if $d(x,y)=2$. 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 give exact formulas of $\lambda(G\cup H)$ and $\lambda(G+H)$. We also prove that $\lambda(G)\le\Delta^2+\Delta$ for any graph $G$ of maximum degree $\Delta$. For odd-sun-free (OSF)-chordal graphs, the upper bound can be reduced to $\lambda(G)\le 2\Delta+1$. For sun-free (SF)-chordal graphs, the upper bound can be reduced to $\lambda(G)\le \Delta+2\chi(G)-2$. Finally, we present a polynomial time algorithm to determine $\lambda(T)$ for a tree $T$.