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, horizontal ellipsis ,b such that the induced mapping f+:V?Zv given by f+x= n-ary sumation x is an element of AA is an element of BfAmod v is a bijection. A quick observation shows that every v,b,r,k,?-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 equivalent to 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,?-BIBDs. We consider the case v equivalent to 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 equivalent to 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.