The neighbour sum distinguishing relaxed edge colouring

A k-edge colouring (not necessarily proper) of a graph with colours in { 1 , 2 , . . . , k } is neighbour sum distinguishing if, for any two adjacent vertices, the sums of the colours of the edges incident with each of them are distinct. The smallest value of k such that such a colouring of G exists is denoted by chi(e)(sigma)(G ) . When we add the additional restriction that the k-edge colouring must be proper, then the smallest value of k such that such a colouring exists is denoted by chi(sigma)'(G). Such colourings are studied on a connected graph on at least 3 vertices. There are two famous conjectures on these edge colourings: the 1-2-3 Conjecture states that chi(sigma)e & nbsp;(G ) <= 3 for any graph G ; and the other states that chi(sigma)'(G) <= delta(G) + 2 for any graph G &NOTEQUexpressionL; C-5 . In this paper, we generalize these two versions of neighbour sum distinguishing edge colourings by introducing the edge colouring in which each monochromatic set of edges induces a subgraph with maximum degree at most d. We call such an edge colouring that distinguishes adjacent vertices a neighbour sum distinguishing d-relaxed k-edge colouring. We denote by chi(sigma)'(d) (G ) the smallest value of k such that such a colouring of G exists. We study families of graphs for which chi(sigma)'& nbsp;is known. We show that the number of required colours decreases when the proper condition is relaxed. In particular, we prove that chi(sigma)'2 (G ) <= 4 for every subcubic graph. For complete graphs, we show that chi(sigma)'d (K-n) <= 4 if d is an element of { n-1/2, . . . , n -1 } and we also determine the exact value of chi(sigma)'2 (K-n). Finally, we determine the value of chi(sigma)'d (T ) for any tree T . (C)& nbsp;2021 Elsevier Inc. All rights reserved.