## Taiwanese Journal of Mathematics

### 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.

#### Article information

Source
Taiwanese J. Math., Volume 15, Number 3 (2011), 1277-1285.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500406299

Digital Object Identifier
doi:10.11650/twjm/1500406299

Mathematical Reviews number (MathSciNet)
MR2829911

Zentralblatt MATH identifier
1235.05119

Subjects
Primary: 05C

#### Citation

Chang, Gerard J.; Lu, Changhong. The $L(2,1)$-$\cal F$-Labeling Problem of Graphs. Taiwanese J. Math. 15 (2011), no. 3, 1277--1285. doi:10.11650/twjm/1500406299. https://projecteuclid.org/euclid.twjm/1500406299

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