On the Total Edge Irregularity Strength of Odd and Even Staircase Graphs

A total \(m\)-labelling \(\alpha\) on graph \(\Gamma\) is called an edge irregular total \(m\)-labelling if for any two different edges of \(\Gamma\), their weights respect to \(\alpha\) are distinct, where the weight of any edge is defined as the sum of its label and the labels of its end vertices. We determine the minimum \(m\) such that \(\Gamma\) can be labelled by an edge irregular total \(m\)-labelling whenever \(\Gamma\) is an odd staircase graph or an even staircase graph.