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Distance-two labelings of digraphs

G. J. Chang·J. -J. Chen·D. Kuo·S. C. Liaw
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Abstract

For positive integers $j\ge k$, an $L(j,k)$-labeling of a digraph $D$ is a function $f$ from $V(D)$ into the set of nonnegative integers such that $|f(x)-f(y)|\ge j$ if $x$ is adjacent to $y$ in $D$ and $|f(x)-f(y)|\ge k$ if $x$ is of distant two to $y$ in $D$. Elements of the image of $f$ are called labels. The $L(j,k)$-labeling problem is to determine the $\vecλ_{j,k}$-number $\vecλ_{j,k}(D)$ of a digraph $D$, which is the minimum of the maximum label used in an $L(j,k)$-labeling of $D$. This paper studies $\vecλ_{j,k}$- numbers of digraphs. In particular, we determine $\vecλ_{j,k}$- numbers of digraphs whose longest dipath is of length at most 2, and $\vecλ_{j,k}$-numbers of ditrees having dipaths of length 4. We also give bounds for $\vecλ_{j,k}$-numbers of bipartite digraphs whose longest dipath is of length 3. Finally, we present a linear-time algorithm for determining $\vecλ_{j,1}$-numbers of ditrees whose longest dipath is of length 3.

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BibTeXRIS

G. J. Chang, J. -J. Chen, D. Kuo, S. C. Liaw. 2004-07-09. Distance-two labelings of digraphs. https://arxiv.org/abs/math/0407167

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