Block-Graceful Designs

In this article, we adapt the edge-graceful graph labeling definition into block designs and define a block design V,B with V=v and B=b as block-graceful if there exists a bijection f:B⟶1,2,…,b such that the induced mapping f+:V⟶Zv given by f+x=∑x∈AA∈BfAmod v is a bijection. A quick observation shows that every v,b,r,k,λ−BIBD that is generated from a cyclic difference family is block-graceful when v,r=1. As immediate consequences of this observation, we can obtain block-graceful Steiner triple system of order v for all v≡1mod 6 and block-graceful projective geometries, i.e., qd+1−1/q−1,qd−1/q−1,qd−1−1/q−1−BIBDs. In the article, we give a necessary condition and prove some basic results on the existence of block-graceful v,k,λ−BIBDs. We consider the case v≡3mod 6 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≡3mod 6. We also consider affine geometries and prove that for every integer d≥2 and q≥3, where q is an odd prime power or q=4, there exists a block-graceful qd,q,1−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.