The Graceful Tree Conjecture claims that every finite simple tree of order n can be vertex labeled with integers {1, 2, ...n} so that the absolute values of the differences of the vertex labels of the end-vertices of edges are all distinct. That is, a graceful labeling of a tree is a vertex labeling f , a bijection f : V (Tn) −→ {1, 2, ...n}, that induces an edge labeling g(uv) = |f(u)− f(v)| t...

The product version of the 1-2-3 Conjecture, introduced by Skowronek-Kaziów in 2012, states that, a few obvious exceptions apart, all graphs can be 3-edge-labelled so that no two adjacent vertices get incident to same labels. To date, this conjecture was mainly verified for complete and 3-colourable graphs. As strong support conjecture, it also proved admit such 4-labellings. In work, we invest...

Graceful tree conjecture is a well-known open problem in graph theory. Here we present a computational approach to this conjecture. An algorithm for finding graceful labelling for trees is proposed. With this algorithm, we show that every tree with at most 35 vertices allows a graceful labelling, hence we verify that the graceful tree conjecture is correct for trees with at most 35 vertices.

A graceful labelling of an undirected graph G with n edges is a one-to-one function from the set of vertices of G to the set {0, 1, 2, . . . , n} such that the induced edge labels are all distinct. An induced edge label is the absolute value of the difference between the two end-vertex labels. The Graceful Tree Conjecture states that all trees have a graceful labelling. In this survey we presen...

In this document, we describe the concepts of ideal semantics and stage semantics for abstract argumentation in terms of argument labellings. The difference between the traditional extensions approach and the labelling approach is that where the former only identifies the sets of accepted arguments, the latter also identifies the rejected arguments as well as the arguments that are neither acce...

We prove the existence of infinite classes of cyclic Γ-decompositions of the complete multipartite graph, Γ being a caterpillar, a hairy cycle or a cycle. All results are obtained by the construction of d-divisible α-labelings of Γ, introduced in [A. Pasotti, On d-graceful labelings, Ars Combin. 111 (2013), 207–223] as a generalization of classical α-labelings. It is known that such labelings i...

Motivated by brane cosmology we solve the Einstein equations with a time dependent cosmological constant. Assuming that at an early epoch the vacuum energy scales as 1/logt, we show that the universe passes from a fast growing phase (inflation) to an expanding phase in a natural way.