Edge irregular total labellings for graphs of linear size

As an edge variant of the well-known irregularity strength of a graph G = ( V , E ) we investigate edge irregular total labellings, i.e. functions f : V ∪ E → { 1 , 2 , … , k } such that f ( u ) + f ( u v ) + f ( v ) ≠ f ( u ′ ) + f ( u ′ v ′ ) + f ( v ′ ) for every pair of different edges u v , u ′ v ′ ∈ E . The smallest possible k is the total edge irregularity strength of G . Confirming a conjecture by Ivančo and Jendrol’ for a large class of graphs we prove that the natural lower bound k = ⌈ m + 2 3 ⌉ is tight for every graph of order n , size m and maximum degree Δ with m > 111000 Δ . This also implies that the probability that a random graph from G ( n , p ( n ) ) satisfies the Ivančo–Jendrol’ Conjecture tends to 1 as n → ∞ for all functions p ∈ [ 0 , 1 ] N . Furthermore, we prove that k = ⌈ m 2 ⌉ is an upper bound for every graph G of order n and size m ≥ 3 whose edges are not all incident to a single vertex.