Total Edge Irregularity Strength of Large Graphs

Let $m:=\mathrm{\mid }E(G)\mathrm{\mid }$ sufficiently large and $s:=(m-1)\mathrm{/}3$. We show that unless the maximum degree $\mathrm{\Delta }>2s$, there is a weighting $w:E\cup V\to \{0,1,\mathrm{.}\mathrm{.}\mathrm{.},s\}$ so that $w(uv)+w(u)+w(v)\mathrm{\ne }w(u^{\mathrm{\prime }}{v}^{\mathrm{\prime }})+w(u^{\mathrm{\prime }})+w(v^{\mathrm{\prime }})$ whenever $uv\mathrm{\ne }{u}^{\mathrm{\prime }}{v}^{\mathrm{\prime }}$ (such a weighting is called {\em total edge irregular}). This validates a conjecture by Ivanco and Jendrol’ for large graphs, extending a result by Brandt, Miskuf and Rautenbach.