## Taiwanese Journal of Mathematics

### ON DISTANCE TWO LABELLING OF UNIT INTERVAL GRAPHS

#### Abstract

An $L(2,1)$-labelling of a graph $G$ is an assignment of non-negative integers to the vertices of $G$ such that vertices at distance at most two get different numbers and adjacent vertices get numbers which are at least two apart. The $L(2,1)$-labelling number of $G$, denoted by $\lambda(G)$, is the minimum range of labels over all such labellings. In this paper, we first discuss some necessary and sufficient conditions for unit interval graph $G$ to have $\lambda(G)=2\chi(G)-2$ and then characterize all unit interval graphs $G$ of order no more than $3\chi(G)-1$, where $\chi(G)$ is the chromatic number of $G$. Finally, we discuss some subgraphs of unit interval graphs $G$ on more than $2\chi(G)+1$ vertices with $\lambda(G) = 2\chi(G)$.

#### Article information

Source
Taiwanese J. Math., Volume 13, Number 4 (2009), 1167-1179.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500405499

Digital Object Identifier
doi:10.11650/twjm/1500405499

Mathematical Reviews number (MathSciNet)
MR2543734

Zentralblatt MATH identifier
1188.05118

Subjects
Primary: 05C78: Graph labelling (graceful graphs, bandwidth, etc.)

#### Citation

Lam, Peter Che Bor; Wang, Tao-Ming; Shiu, Wai Chee; Gu, Guohua. ON DISTANCE TWO LABELLING OF UNIT INTERVAL GRAPHS. Taiwanese J. Math. 13 (2009), no. 4, 1167--1179. doi:10.11650/twjm/1500405499. https://projecteuclid.org/euclid.twjm/1500405499

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