## Taiwanese Journal of Mathematics

### NEW $L(j,k)$-LABELINGS FOR DIRECT PRODUCTS OF COMPLETE GRAPHS

#### Abstract

An $L(j,k)$-$labeling$ of a graph is a vertex labeling such that the difference between the labels of adjacent vertices is at least $j$ and that between vertices separated by a distance 2 is at least $k$. The minimum of the spans of all $L(j,k)$-labelings of $G$ is denoted by $\lambda_k^j(G)$. Recently, Haque and Jha [16] proved that if $G$ is a multiple direct product of complete graphs, then $\lambda_k^j(G)$ coincides with the trivial lower bound $(N-1)k$, where $N$ is the order of $G$ and $\frac{j}{k}$ is within a certain bound. In this paper, we suggest a new labeling method for such a graph $G$. With this method, we extend the range of $\frac{j}{k}$ such that $\lambda_k^j(G)= (N-1)k$ holds. Moreover, we obtain the upper bound of $\lambda_k^j(G)$ for the remaining cases in the range $\frac{j}{k}$.

#### Article information

Source
Taiwanese J. Math., Volume 18, Number 3 (2014), 793-807.

Dates
First available in Project Euclid: 10 July 2017

https://projecteuclid.org/euclid.twjm/1499706441

Digital Object Identifier
doi:10.11650/tjm.18.2014.3045

Mathematical Reviews number (MathSciNet)
MR3213387

Zentralblatt MATH identifier
1357.05135

#### Citation

Kim, Byeong Moon; Song, Byung Chul; Rho, Yoomi; Hwang, Woonjae. NEW $L(j,k)$-LABELINGS FOR DIRECT PRODUCTS OF COMPLETE GRAPHS. Taiwanese J. Math. 18 (2014), no. 3, 793--807. doi:10.11650/tjm.18.2014.3045. https://projecteuclid.org/euclid.twjm/1499706441

#### References

• T. Calamoneri, The $L(h,k)$-labeling problem: A survey and annotated bibliography, Comp. J., 49 (2006), 585-608.
• T. Calamoneri and R. Petreschi, The $L(h,k)$-labeling problem: an updated survey and annotated bibliography, Comp. J., 54 (2011), 1344-1371.
• T. Calamoneri, $L(h,1)$-labeling subclasses of planar graphs, J. Parallel Distrib. Comp., 54 (2011), 1344-1371.
• T. Calamoneri, E. G. Fusco, R. B. Tan and P. Vocca, $L(h,1,1)$-labeling of outerplanar graphs, Math. Methods Oper. Res., 69 (2009), 307-321.
• G. J. Chang and C. Lu, Distance-two labelings of graphs, Europ. J. Combin., 24 (2003), 53-58.
• G. J. Chang and D. Kuo, The $L(2,1)$-labeling problem on graphs, SIAM J. Disc. Math., 9 (1996), 309-316.
• G. J. Chang, W. Ke, D. Kuo, D. Liu and R. K. Yeh, On $L(d,1)$-labelings of graphs, Disc. Math., 220 (2000), 57-66.
• G. Chartrand, D. Erwin and P. Zhang, A graph labeling problem suggested by FM channel restrictions, Bull. ICA, 43 (2005), 43-57.
• M. L. Chia, D. Kuo, H. Liao, C. H. Yang and R. K. Yeh, $L(3,2,1)$ labeling of graphs, Taiwanese J. Math., 15 (2011), 2439-2457.
• S. H. Chiang and H. H. Yan, On $L(d,1)$-labeling of Cartesian product of a cycle and a path, Disc. Appl. Math., 156(15) (2008), 2867-2881.
• J. P. Georges and D. W. Mauro, Generalized vertex labelings with a condition at distance two, Congr. Numer., 109 (1995), 141-159.
• J. P. Georges and D. W. Mauro, Labeling trees with a condition at distance two, Disc. Math., 269 (2003), 127-148.
• J. P. Georges, D. W. Mauro and M. I. Stein, Labeling products of complete graphs with a condition at a distance two, SIAM J. Disc. Math., 14 (2000), 28-35.
• J. R. Griggs and R. K. Yeh, Labeling graphs with a condition at distance two, SIAM J. Disc. Math., 5 (1992), 586-595.
• W. K. Hale, Frequency assignment: theory and applications, Proc. IEEE, 68 (1980), 1497-1514.
• E. Haque and P. K. Jha, $L(j,k)$-labelings of Kronecker products of complete graphs, IEEE Trans. Circuits Syst. II: Express Briefs, 55(1) (2008), 70-73.
• F. Havet, B. Reed and J. S. Sereni, Griggs and Yeh's Conjecture and $L(p,1)$-labelings, SIAM J. Disc. Math., 26(1) (2012), 145-168.
• P. K. Jha, Optimal $L(2,1)$-labeling of strong products of cycles, IEEE Trans. Circuits Syst. I: Fund. Theory Appl., 48 (2001), 498-500.
• P. K. Jha, S. Klavžar and A. Vesel, Optimal $L(d,1)$-labelings of certain direct products of cycles and Cartesian product of cycles, Disc. Appl. Math., 152(1-3) (2005), 257-265.
• J. H. Kang, $L(2,1)$-labeling of Hamiltonian graphs with maximum degree $3$, SIAM J. Disc. Math., 22(1) (2008), 213-230.
• B. M. Kim, B. C. Song and W. Hwang, Distance three labelings for direct products of 3 complete graphs, Taiwanese J. of Math., 17 (2013), 207-219.
• B. M. Kim, B. C. Song and W. Hwang, Distance three labelings for $K_n \times K_2$, Intern. J. Comp. Math., 90 (2013), 906-911.
• B. M. Kim, B. C. Song and Y. Rho, $L(2,1)$-labelings for ditect products of a triangle and a cycle, Intern. J. Comp. Math., 90 (2013), 475-482.
• S. Klavzar and A. Vesel, Computing graph invariants on rotagraphs using dynamic algorithm approach: the case of $(2,1)$-colorings and independence numbers, Disc. Appl. Math., 129 (2003), 449-460.
• S. R. Kola and P. Panigrahi, An improved lower bound for the radio $k$-chromatic number of the hypercube $Q_n$, Computers & Math. with Appl., 60 (2010), 2131-2140.
• D. Kuo and J. H. Yan, On $L(2,1)$-labelings of Cartesian products of paths and cycles, Disc. Math., 283 (2004), 137-144.
• P. C. B. Lam, W. Lin and J. Wu, $L(j,k)$- and circular $L(j,k)$-labelings for the products of complete graphs, J. Comp. Optim., 14 (2007), 219-227.
• W. Lin and P. C. B. Lam, Distance two labeling and direct products of graphs, Disc. Math., 308 (2008), 3805-3815.
• D. D. Liu, Radio number for trees, Disc. Math., 308(7) (2008), 1153-1164.
• D. D. Liu and M. Xie, Radio number for square paths, Ars. Combinatoria, 90 (2009), 307-319.
• D. D. Liu and X. Zhu, Multilevel distance labelings for paths and cycles, SIAM J. Disc. Math., 19 (2005), 610-621.
• D. Sakai, Labeling chordal graphs with a condition at distance two, SIAM J. Disc. Math., 7 (1994), 133-140.
• C. Schwarz and D. S. Troxell, $L(2,1)$-labelings of Cartesian products of two cycles, Disc. Appl. Math., 154(10) (2006), 1522-1540.
• M. A. Whittlesey, J. P. Georges and D. W. Mauro, On the $\lambda$-number of $Q_n$ and related graphs, SIAM J. Disc. Math., 8 (1995), 499-506.
• R. K. Yeh, A survey on labeling graphs with a condition at distance two, Disc. Math., 306(12) (2006), 1217-1231.