Distant irregularity strength of graphs

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