LINE-GRACEFUL DESIGNS - Erdemir, D.; Kolotoglu, E. (COMENIUS UNIV, SCH MEDICINE, 2024)
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, B) with |V| = v as line-graceful if there exists a function f : B -> {0, 1, ... , v - 1} such that the induced mapping f(+): V -> Z(v) given by f+(x) = Sigma(A is an element of B) : (x is an element of A) f(A) (mod 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 2n for all n >= 3 by using a recursive construction.
Keywords: Line-graceful design; Steiner triple system; Steiner quadruple system; affine geometry; projective geometry; Mathematics