On the Irregularity Strength of Dense Graphs

Note on the group edge irregularity strength of graphs

“…The smallest value of s that allows an irregular labeling is called the irregularity strength of G and denoted by s(G). This problem was one of the major sources of inspiration in graph theory [3,4,5,6,7,12,18,19,20,23,26,28]. For example the concept of G-irregular labeling is a generalization of irregular labeling on Abelian groups.…”

Distant total sum distinguishing index of graphs

“…, k} exists with w∈N (u) c(uw) = w∈N (v) c(vw) for every u, v ∈ V , u = v. This graph invariant has been analyzed in multiple papers, see e.g. [3,7,12,13,14,16,19,22,26,28,33], but also gave rise to a fast-developing branch of research, which may be referred to as "additive graph labellings", or more generally "vertex distinguishing colourings", see [1,2,6,8,9,15,20,29,35,36,37] for examples of papers introducing a few representative concepts of this branch. One of these is a problem of a total neighbour sum distinguishing colouring of a given graph graph G = (V, E), i.e.…”

Distant sum distinguishing index of graphs with bounded minimum degree

“…This we shall also call the sum at v, see e.g. [3,5,9,10,12,13,15,17,21,22,24,30,31] for a few out of a vastness of results concerning s(G), which also gave rise to a whole discipline devoted to investigating this and other related problems. One of the most intriguing direct descendants of the irregularity strength is its local correspondent, where we necessarily require an inequality d c (u) = d c (v) to hold only for adjacent vertices u, v in G. The least k admitting a colouring c : E → {1, 2, .…”