Block-Graceful Designs

Abstract

In this article, we adapt the edge-graceful graph labeling definition into block designs and define a block design $\left(V,B\right)$ with $\left|V\right|=v$ and $\left|B\right|=b$ as block-graceful if there exists a bijection $f:B⟶\left\{\mathrm{1,2},\dots ,b\right\}$ such that the induced mapping $f^{+}:V⟶{\mathbb{Z}}_v$ given by $f^{+}\left(x\right)={\sum }_{\begin{array}{c}x\in A\\ A\in B\end{array}}f\left(A\right)\left(\mathrm{mod}v\right)$ is a bijection. A quick observation shows that every $\left(v,b,r,k,\lambda \right)-$ BIBD that is generated from a cyclic difference family is block-graceful when $\left(v,r\right)=1$ . As immediate consequences of this observation, we can obtain block-graceful Steiner triple system of order $v$ for all $v\equiv 1\left(\mathrm{mod}6\right)$ and block-graceful projective geometries, i.e., $\left(\left(q^{d+1}-1\right)/\left(q-1\right),\left(q^d-1\right)/\left(q-1\right),\left(q^{d-1}-1\right)/\left(q-1\right)\right)-$ BIBDs. In the article, we give a necessary condition and prove some basic results on the existence of block-graceful $\left(v,k,\lambda \right)-$ BIBDs. We consider the case $v\equiv 3\left(\mathrm{mod}6\right)$ for Steiner triple systems and give a recursive construction for obtaining block-graceful triple systems from those of smaller order which allows us to get infinite families of block-graceful Steiner triple systems of order $v$ for $v\equiv 3\left(\mathrm{mod}6\right)$ . We also consider affine geometries and prove that for every integer $d\ge 2$ and $q\ge 3$ , where $q$ is an odd prime power or $q=4$ , there exists a block-graceful $\left(q^d,q,1\right)-$ BIBD. We make a list of small parameters such that the existence problem of block-graceful labelings is completely solved for all pairwise nonisomorphic BIBDs with these parameters. We complete the article with some open problems and conjectures.