Vendredi 15 mars 2019, 14hrs, salle 25-26/105, LIP6, UPMC.

4 Place Jussieu, 75005, Paris.

**Abstract**

The main question that we explore was introduced by Karonski, Luczak and Thomason in 2004 : Can we weight the edges of a graph G , with weights 1 ,2 , and 3 , such that any two of adjacent vertices of G are distinguished by the sum of their incident weights ? This question later becomes the famous 1-2-3 Conjecture.

In this presentation we explore several variants of the 1-2-3 Conjecture, and their links with locally irregular decompositions. We are interested in both optimisation results and algorithmic problems. We first introduce an equitable version of the neighbour-sum-distinguishing edge-weightings, that is a variant where we require every edge weight to be used the same number of times up to a difference of 1. After that we explore how neighbour-sum-distinguishing weightings behave if we require sums of neighbouring vertices to differ by at least 2. Namely, we present results on the smallest maximal weight needed to construct such weightings for some classes of graphs, and study some algorithmic aspects of this problem. Due to the links between neighbour-sum-distinguishing edge weightings and locally irregular decompositions, we also explore the locally irregular index of subcubic graphs, along with other variants of the locally irregular decomposition problem. Finally, we present a more general work toward a general theory unifying neighbour-sum-distinguishing edge-weightings and locally irregular decompositions.