The total edge irregularity strength 
of a finite simple
graph 
is the least positive integer 
for which labels from 1, 2, ..., 
can be assigned to both its vertices
and edges so that the weights 
are different for all edges 
(Bača et al. 2007). Repeated labels are permitted. The requirement concerns
the sums associated with edges, not distinctness of
the labels themselves.
This differs from irregularity strength, which labels only edges and distinguishes the induced sums at vertices.
Barra et al. (2026) proved that every cubic graph with 
vertices satisfies
 |
The theorem includes disconnected graphs. Their proof supplies a deterministic labeling construction.
