Total edge irregularity strength of large graphs

Let m ≔ | E ( G ) | sufficiently large and s ≔ ⌈ ( m − 1 ) / 3 ⌉ . We show that unless the maximum degree Δ > 2 s , there is a weighting w ˆ : E ∪ V → { 0 , 1 , … , s } so that w ˆ ( u v ) + w ˆ ( u ) + w ˆ ( v ) ≠ w ˆ ( u ′ v ′ ) + w ˆ ( u ′ ) + w ˆ ( v ′ ) whenever u v ≠ 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.