On a conjecture about edge irregular total labelings

Abstract

Abstract As our main result, we prove that for every multigraph G = ( V, E ) which has no loops and is of order n , size m , and maximum degree $\Delta < {{{10}}^{-{{3}}}{{m}}\over \sqrt{{{8}}{{n}}}}$ there is a mapping ${{f}}:{{V}}\cup {{E}}\to \big\{{{1}},{{2}},\ldots,\big\lceil{{{m}}+{{2}}\over {{3}}}\big\rceil\big\}$ such that ${{f}}({{u}})+{{f}}({{uv}})+{{f}}({{v}})\not={{f}}({{u}}')+{{f}}({{u}}'{{v}}')+{{f}}({{v}}')$ for every ${{uv}},{{u}}'{{v}}'\in {{E}}$ with ${{uv}}\not={{u}}'{{v}}'$ . Functions with this property were recently introduced and studied by Bača et al. and were called edge irregular total labelings. Our result confirms a recent conjecture of Ivančo and Jendrol′ about such labelings for dense graphs, for graphs where the maximum and minimum degree are not too different in terms of the order, and also for large graphs of bounded maximum degree. © 2008 Wiley Periodicals, Inc. J Graph Theory 57: 333–343, 2008