Graph labellings have been a very fruitful area of research in the last four decades. However, despite the staggering number of papers published in the field (over 1000), few general results are available, and most papers deal with particular classes of graphs and methods. Here we approach the problem from the computational viewpoint, and in a quite general way. We present the existence problem...

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.

A (simple undirected) graph G = (V,E) with m edges is graceful if it has a distinct vertex labeling f : V −→ {0, 1, 2, 3, . . . ,m} which induces a set of distinct edge labels {|f(u)− f(v)| | uv ∈ E, u, v ∈ V }. The famous Ringel-Kotzig conjecture [9, 15] is that all trees are graceful. The base of a tree T is obtained from T by deleting its one-degree vertices. A caterpillar is a tree whose ba...

The 1-2-3 Conjecture states that every connected graph different from K2 admits a proper 3-(edge-)labelling, i.e., can have its edges labelled with 1,2,3 so no two adjacent vertices are incident to the same sum of labels. In connection some recent optimisation variants this conjecture, in paper we investigate role label 3 3-labellings graphs. An intuition previous investigations is that, genera...

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...