Line-Graceful Designs

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.