Taiwanese Journal of Mathematics

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

Byeong Moon Kim, Byung Chul Song, Yoomi Rho, and Woonjae Hwang

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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

Permanent link to this document
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

Subjects
Primary: 05C78: Graph labelling (graceful graphs, bandwidth, etc.) 05C12: Distance in graphs 05C76: Graph operations (line graphs, products, etc.)

Keywords
$L(j,k)$-labelings direct product of graphs complete graph channel-assignment problem

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


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