Edge irregular total labellings for graphs of linear size

Confirming a conjecture by Ivančo and Jendrol’ for a large class of graphs we prove that for every graph G = (V,E) of order n, size m and maximum degree ∆ with m > 111000∆ there is a function f : V ∪ E → { 1, 2, ..., ⌈ m+2 3 ⌉} such that f(u) + f(uv) + f(v) 6= f(u′) + f(u′v′) + f(v′) for every uv, u′v′ ∈ E with uv 6= u′v′. Furthermore, we prove the existence of such a function with values up to ⌈ m 2 ⌉ for every graph G = (V,E) of order n and size m ≥ 3 whose edges are not all incident to a single vertex.