# Line-Graceful Designs

## Main Article Content

## Abstract

In [3], the authors adapted the edge-graceful graph labeling definition into block designs. In this article, we adapt the line-graceful graph labeling definition into block designs and define a block design $(V,\mathcal{B})$ with $|V|=v$ as line-graceful if there exists a function $f: \mathcal{B} \rightarrow \{0,1,\dots,v-1\}$ such that the induced mapping $f^{+}: V \rightarrow \mathbb{Z}_{v}$ given by $f^{+}(x)=\sum_{A\in \mathcal{B} : x\in A}{f(A)}\pmod{v}$ is a bijection. In this article, the cases that are incomplete in terms of block-graceful labelings, are completed in terms of line-graceful labelings. Moreover, we prove that there exists a line-graceful Steiner quadruple system of order $2^{n}$ for all $n \geq 3$ by using a recursive construction.

## Article Details

How to Cite

Erdemir, D., & Kolotoğlu, E.
(2024).
Line-Graceful Designs.

*Acta Mathematica Universitatis Comenianae, 93*(3), 129-136. Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/1998/1052
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