In this article, we adapt the edge-graceful graph labeling definition into block designs and define a design V , B with id="M2"> open="|" close="|" = v id="M3"> b as block-graceful if there exists bijection ...

An a-labeling of a bipartite graph G with n edges easily yields both a cyclic G-decomposition of Kn,n and of K2nx+1 for all positive integers x. A ,B-Iabeling (or graceful labeling) of G yields a cyclic decomposition of K2n+1 only. It is well-known that certain classes of trees do not have a-Iabelings. In this article, we introduce the concept of a near a-labeling of a bipartite graph, and prov...

A graceful labeling of a graph with q edges is a labeling of its vertices using the integers in [0, q], such that no two vertices are assigned the same label and each edge is uniquely identified by the absolute difference between the labels of its endpoints. The well known Graceful Tree Conjecture (GTC) states that all trees are graceful, and it remains open. It was proved in 1999 by Broersma a...

We propose a stringy version of the old inflation scenario which does not require any slowroll inflaton potential and is based on a specific example of string compactification with warped metric. Our set-up admits the presence of anti-D3-branes in the deep infrared region of the metric and a false vacuum state with positive vacuum energy density. The latter is responsible for the accelerated pe...

We present several constructions for terraces for cyclic groups that are zigzag terraces, graceful terraces or both. These allow us to complete the classification of imperfect twizzler terraces, unify several previously unrelated constructions of terraces from the literature, find terraces for some non-cyclic abelian groups, and show that for any given r the Oberwolfach Problem with parameters ...

In this paper we study a technique to transform α-labeled trees into ρ-labeled forests. We use this result to prove that the complete graph K2n+1 can be decomposed into these types of forests. In addition we show a robust family of trees that admit ρ-labelings, we use this result to describe the set of all trees for which a ρ-labeling must be found to completely solve Kotzig’s conjecture about ...