Total Edge Irregularity Strength of Large Graphs

TL;DR

This paper proves that for large graphs with certain degree constraints, a total edge irregular weighting exists, confirming a conjecture and extending previous results in graph irregularity theory.

Contribution

It establishes the existence of total edge irregular weightings for large graphs under specific degree conditions, confirming a conjecture by Ivanco and Jendrol' and extending prior work.

Findings

Validates the conjecture for large graphs

Extends previous results by Brandt, Miskuf, and Rautenbach

Provides a constructive method for total edge irregular weightings

Abstract

Let $m:=|E(G)|$ sufficiently large and $s:=(m-1)/3$. We show that unless the maximum degree $\Delta > 2s$, there is a weighting $w:E\cup V\to \{0,1,...,s\}$ so that $w(uv)+w(u)+w(v)\ne w(u'v')+w(u')+w(v')$ whenever $uv\ne u'v'$ (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.