Block-Graceful Designs

Abstract

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 ⟶ Z v given by f + x = ∑ x ∈ A A ∈ B f A mod 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 ≡ 1 mod 6 and block-graceful projective geometries, i.e., q d + 1 − 1 / q − 1 , q d − 1 / q − 1 , q d − 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 ≡ 3 mod 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 ≡ 3 mod 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 q d , 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.