Taiwanese Journal of Mathematics

ON DISTANCE TWO LABELLING OF UNIT INTERVAL GRAPHS

Peter Che Bor Lam, Tao-Ming Wang, Wai Chee Shiu, and Guohua Gu

Full-text: Open access

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

Permanent link to this document
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.)

Keywords
$L(2,1)$-labelling unit interval graph

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


Export citation

References

  • G. J. Chang, W.-T. Ke, D. Kuo, D. D.-F. Liu and R. K. Yeh, On $L(d,1)$-labellings of graphs, Discrete Math., 220 (2000), 57-66.
  • G. J. Chang and D. Kuo, The $L(2,1)$-labelling problem on graphs, SIAM J. Discrete Math., 9 (1996), 309-316
  • J. P. Georges, D. W. Mauro and M. I. Stein, Labelling products of complete graphs with a condition at distance two, SIAM J. Discrete Math., 14 (2000), 28-35.
  • J. R. Griggs and R. K. Yeh, Labelling graphs with a condition at distance two, SIAM J. Discrete Math., 5 (1992), 586-595.
  • W. K. Hale, Frequency Assignment: Theory and Applications, Proc. IEEE, 68 (1980), 1497-1514.
  • J. van den Heuvel, R. A. Leese and M. A. Shepherd, Graph labelling and radio channel assignment, J. Graph Theory, 29 (1988), 263-283.
  • D. D.-F. Liu and R. K. Yeh, On distance two labellings of graphs, Ars Combinatoria, 47 (1997), 13-22.
  • F. S. Roberts (1971), On the compatibility between a graph and a simple order, J. Combin. Theory, 11 (1971), 28-38.
  • D. Sakai, Labeling chordal graphs with a condition at distance two, SIAM J. Discrete Math., 7 (1994), 133-140.
  • M. A. Whittlesey, J. P. Georges and D. W. Mauro, On the $\l$-number of $Q_n$ and related graphs, SIAM J. Discrete Math., 8 (1995), 499-506.
  • K.-F. Wu and R. K. Yeh, Labelling graphs with the circular difference, Taiwanese J. Math., 4 (2000), 397-405.