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 study the diagrams of affine permutations and their balanced labellings. As in the finite case, which was investigated by Fomin, Greene, Reiner, and Shimozono, the balanced labellings give a natural encoding of reduced decompositions of affine permutations. In fact, we show that the sum of weight monomials of the column strict balanced labellings is the affine Stanley symmetric function defi...

The additive hereditary property of linear forests is characterized by the existence of average labellings.

Probabilistic argumentation combines the quantitative uncertainty accounted by probability theory with the qualitative uncertainty captured by argumentation. In this paper, we investigate the problem of learning the structure of an argumentative graph to account for (a distribution of) labellings of a set of arguments. We consider a general abstract framework, where the structure of arguments i...

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...

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 ...