Taiwanese Journal of Mathematics


Wensong Lin and Jianzhuan Wu

Full-text: Open access


Let $m$, $j$ and $k$ be positive integers with $j \ge k$. An $m$-circular-$L(j,k)$-edge-labeling of a graph $G$ is an assignment $f$ from $\{0,1,\dots,m-1\}$ to the edges of $G$ such that, for any two edges $e_1$ and $e_2$, $|f(e_1)-f(e_2)|_m\geq j$ if $e_1$ and $e_2$ are adjacent, and $|f(e_1)-f(e_2)|_m \geq k$ if $e_1$ and $e_2$ are at distance $2$, where $|a|_m = \min \{a,m-a\}$. The minimum $m$ such that $G$ has an $m$-circular-$L(j,k)$-edge-labeling is defined as the circular-$L(j,k)$-edge-labeling number of $G$, denoted by $\sigma_{j,k}'(G)$. This paper determines the circular-$L(2,1)$-edge-labeling numbers of the infinite $\Delta$-regular tree for $\Delta \ge 2$ and the $n$-dimensional cube for $n \in \{2,3,4,5\}$.

Article information

Taiwanese J. Math., Volume 16, Number 6 (2012), 2063-2075.

First available in Project Euclid: 18 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C15: Coloring of graphs and hypergraphs

circular-$L(j,k)$-edge-labeling number $\Delta$-regular tree $n$-dimensional cube


Lin, Wensong; Wu, Jianzhuan. ON CIRCULAR-$L(2,1)$-EDGE-LABELING OF GRAPHS. Taiwanese J. Math. 16 (2012), no. 6, 2063--2075. doi:10.11650/twjm/1500406839. https://projecteuclid.org/euclid.twjm/1500406839

Export citation


  • T. Calamoneri, The $L(h,k)$-labelling problem: a survey and annotated bibliography, The Computer Journal, 49(5) (2006), 585-608.
  • Q. Chen and W. Lin, $L(j,k)$-Labelings and $L(j,k)$-edge-Labelings of graphs, to appear in Ars Combin., 2007.
  • J. P. Georges and D. W. Mauro, Edge labelings with a condition at distance two, Ars Combin., 70 (2004), 109-128.
  • J. R. Griggs and X. T. Jin, Recent progress in mathematics and engineering on optimal graph labellings with distance conditions, J. of Combin. Optim., 14(2-3) (2007), 249-257.
  • J. R. Griggs and R. K. Yeh, Labelling graphs with a condition at distance $2$, SIAM J. Discrete Math., 5 (1992), 586-595.
  • W. K. Hale, Frequency assignment: theory and applications, Proc. IEEE, 68 (1980), 1497-1514.
  • J. Heuvel, R. A. Leese and M. A. Shepherd, Graph labeling and radio channel assignment, J. Graph Theory, 29 (1998), 263-283.
  • P. C. B. Lam, W. Lin and J. Wu, $L(j,k)$- and circular $L(j,k)$-labellings for the products of complete graphs, J. Combin. Optim., 14 (2007), 219-227.
  • D. D. F. Liu, Hamiltonicity and circular distance two labellings, \em Discrete Math., 232 (2001), 163-169.
  • D. L$\ddot{u}$, W. Lin, and Z. Song, $L(2,1)$-circular labelings of Cartesian products of complete graphs, J. Mathematical Research & Exposition, 29(1) (2009), 91-98.
  • R. A. Leese and S. D. Noble, Cyclic labellings with constraints at two distance, Electronic J. Combin., 11 (2004), $\sharp$R16.
  • D. D. F. Liu and X. Zhu, Circulant Distant Two Labeling and Circular Chromatic Number, Ars Combin., 69 (2003), 177-183.
  • K. Wu and R. K. Yeh, Labelling graphs with the circular difference, Taiwanese J. Math., 4 (2000), 397-405.
  • R. K. Yeh, A survey on labeling graphs with a condition at distance two, Discrete Math., 306 (2006), 1217-1231.