## Taiwanese Journal of Mathematics

### T-COLORING ON FOLDED HYPERCUBES

#### Abstract

Given a graph $G=(V,E)$ and a set $T$ of non-negative integers containing $0$, a $T$-coloring of $G$ is an integer function $f$ of the vertices of $G$ such that $|f(u)-f(v)| \notin T$ whenever $uv \in E$. The edge-span of a $T$-coloring $f$ is the maximum value of $|f(u)-f(v)|$ over all edges $uv$, and the $T$-edge-span of a graph $G$ is the minimum value of the edge-span among all possible $T$-colorings of $G$. This paper discusses the $T$-edge span of the folded hypercube network of dimension $n$ for the $k$-multiple-of-$s$ set, $T=\{ 0$, $s$, $2s$, $\dots$, $ks\} \cup S$, where $s$ and $k \geq 1$ and $S \subseteq \{ s+1, s+2, \dots , ks-1 \}$.

#### Article information

Source
Taiwanese J. Math., Volume 13, Number 4 (2009), 1331-1341.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500405511

Digital Object Identifier
doi:10.11650/twjm/1500405511

Mathematical Reviews number (MathSciNet)
MR2543746

Zentralblatt MATH identifier
1201.05038

Subjects
Primary: 05C15: Coloring of graphs and hypergraphs

#### Citation

Juan, Justie Su-Tzu; Sun, I-fan; Wu, Pin-Xian. T-COLORING ON FOLDED HYPERCUBES. Taiwanese J. Math. 13 (2009), no. 4, 1331--1341. doi:10.11650/twjm/1500405511. https://projecteuclid.org/euclid.twjm/1500405511

#### References

• G. J. Chang, D. D.-F. Liu, Xuding Zhu, Distance graphs and $T$-coloring, J. Combin. Theory, Ser. B, 75(2) (1999), 259-269.
• M. B. Cozzens and F. S. Roberts, $T$-Colorings of graphs and the channel assignment problem, Congressus Numerantium, 35 (1982), 191-208.
• A. El-Amawy and S. Latifi, Properties and performance of folded hypercubes, IEEE Trans. on Parallel Distributed Syst., 2(1) (1991), 31-42.
• Xinmin Hou, Min Xu and Jun-Ming Xu, Forwarding indices of folded $n$-cubes, Disc. Appl. Math., 145(3) (2005), 490-492.
• S.-J. Hu, S.-T. Juan and G. J. Chang, $T$-colorings and $T$-edge spans of graphs, Graphs and Combinatorics, 15 (1999), 295-301.
• R. Janczewski, Divisibility and $T$-span of graphs, Disc. Math., 234(1-3) (2001), 171-179.
• C.-N. Lai, G.-H. Chen and D.-R. Duh, Constructing one-to-many disjoint paths in folded hypercubes, IEEE Trans. on Computers, 51(1) (2002), 33-45.
• S. Lakshmivarahan, J.-S. Jwo and S. K. Dhall, Symmertry in interconnection networks based on Cayley graphs of permutation groups: A survey, Parallel Comput., 19 (1993), 361-407.
• D. D.-F. Liu, $T$-graphs and the channel assignment problem, Disc. Math., 161(1-3) (1996), 197-205.
• D. D.-F. Liu and R. Yeh, Graph homomorphism and no-hole $T$-coloring, Congressus Numerantium, 138 (1999), 39-48.
• A. Raychaudhuri, Intersection assignments, $T$-coloring, and powers of graphs, Ph.D. Thesis, Department of Mathematics, Rutgers University, New Brunswich, NJ, 1985.
• A. Raychaudhuri, Further results on $T$-coloring and Frequency assignment problems, SIAM J. Disc. Math., 7(4) (1994), 605-613.
• F. S. Roberts, $T$-colorings of graphs: recent results and open problems, Disc. Math., 93 (1991), 229-245.
• B. A. Tesman, $T$-colorings, list $T$-colorings, and set $T$-colorings of graphs, Ph.D. Thesis, Department of Mathematics, Rutgers University, New Brunswich, NJ, 1989.
• B. A. Tesman, List $T$-colorings of graphs, Disc. Appl. Math., 45 (1993), 277-289.
• D. Wang, Embedding hamiltonian cycles into folded hypercubes with faulty links, J. of Parallel and Distributed Computing, 61(4) (2001), 545-564.
• J.-M. Xu and M. Ma, Cycles in folded hypercubes, Appl. Math. Lett., 19(I.2), (2006), 140-145.