The 1-2-3 conjecture asserts that every finite simple graph with no connected component consisting
of a single graph edge admits edge
weights from 
such that adjacent vertices have different
sums of incident edge weights.
It was proposed by Karoński et al. (2004) and proved by Keusch (2024).
For 
and 
,
define
 |
The required condition is
 |
for every graph edge 
. Thus 
gives a vertex coloring,
although the assigned edge weights need not give an edge coloring.
The exclusion of isolated edges is necessary, since both endpoints of such an graph edge receive the same
sum. Isolated vertices
are allowed. The triangle graph requires three
distinct edge weights, so replacing 
by 
does not suffice in general.
The condition is local, whereas irregularity strength requires different incident-weight sums at all vertices, including nonadjacent pairs (Keusch 2024).
