Total edge irregularity strength of large graphs

Let m≔|E(G)|m≔|E(G)| sufficiently large and s≔⌈(m−1)/3⌉s≔⌈(m−1)/3⌉. We show that unless the maximum degree Δ>2sΔ>2s, there is a weighting wˆ:E∪V→{0,1,…,s} so that wˆ(uv)+wˆ(u)+wˆ(v)≠wˆ(u′v′)+wˆ(u′)+wˆ(v′) whenever uv≠u′v′uv≠u′v′ (such a weighting is called total edge irregular). This validates a conjecture by Ivančo and Jendrol’ for large graphs, extending a result by Brandt, Miškuf and Rautenbach.