Distant irregularity strength of graphs

Let G=(V,E)G=(V,E) be a graph, and let c:V→{1,2,…,k}c:V→{1,2,…,k} be a not necessarily proper edge colouring. The weight, or the weighted degree  , of v∈Vv∈V is then defined as w(v)=∑u∈N(v)c(vu)w(v)=∑u∈N(v)c(vu). The colouring cc is said to be irregular if w(u)≠w(v)w(u)≠w(v) for every two distinct vertices u,v∈Vu,v∈V. The smallest kk for which such a colouring exists is called the irregularity strength   of a graph, denoted by s(G)s(G).In this paper we further develop the study of irregular colourings, and require that the colouring cc provides distinct weights only for vertices at distance at most rr. The corresponding parameter is then called the rr-distant irregularity strength  , and denoted by sr(G)sr(G). This notion binds the known 1-2-3 Conjecture posed by Karoński Łuczak and Thomason, whose objective is s1(G)s1(G), with the irregularity strength, as it is justified to write s(G)=s∞(G)s(G)=s∞(G) in this context. We prove that for each positive integer rr, sr(G)≤6Δr−1sr(G)≤6Δr−1.We also investigate a total version of the problem, where given a colouring c:V∪E→{1,2,…,k}c:V∪E→{1,2,…,k} of GG, we define t(v)=c(v)+∑u∈N(v)c(vu)t(v)=c(v)+∑u∈N(v)c(vu) for v∈Vv∈V. The smallest kk for which such a colouring cc exists with t(u)≠t(v)t(u)≠t(v) for every pair of distinct vertices at distance at most rr in GG is called the rr-distant total irregularity strength   of GG, and denoted by tsr(G). We prove that tsr(G)≤3Δr−1 and we discuss that the bounds obtained for both problems are of the right magnitude.This direction of research is inspired by the concept of distant chromatic numbers. The results obtained are also strongly related with the study on the Moore bound.