Locally Irregular Decompositions
This document presents conjectures related to distinguishing adjacent vertices in a graph. The first one, named 1-2-3 Conjecture, asks if it is possible to assign a weight of 1, 2 or 3 to each edge of a graph G such that, for each pair of adjacent vertices u and v in V(G), the sum of the weights of the edges that are incident to u and v is different. The second conjecture asks if it is possible to color the edges of a graph with at most 3 colors such that the subgraph induced by each color has no adjacent vertices with an equal degree. This research project presents the main results in the literature, and suggests some directions for future research. It also presents some of our results in the second conjecture.