Let G be a graph. A graph is nice if none of its connected components is isomorphic to K_2, and let chi_rm S(G) be the smallest k≥ 1 for which G has an s-proper k-labelling, meaning an edge-labelling from 1,…,k whose incident-label sums give different values at the ends of every edge. The 1-2-3 Conjecture. If G is a nice graph, then chi_rm S(G)≤ 3. This is the original sum version of the 1-2-3 Conjecture, raised by Karoński, Łuczak and Thomason. The paper identified by the source is a proof of this conjecture, so the claim is solved.

The 1-2-3 Conjecture for sum-proper edge-labellings

Posed by Karoński, Łuczak, and Thomason in 2004, the conjecture asserts that every nice graph has an edge-labelling with labels $1,2,3$ giving different incident-label sums at adjacent vertices.

Known results

Every nice graph was known to satisfy $\chi_{\rm S}(G)\leq 5$ by 2010.
The conjecture was proved for graphs of maximum average degree less than $8/3$ (2013).
Three labels suffice when the minimum degree satisfies $\delta\geq C\log\Delta$ .
Bensmail, Fioravantes, and Mc Inerney (2021) proved that arbitrarily many edges labelled $3$ may be necessary and studied the associated decision problem.

Claimed solution (date not stated)

The paper A Solution to the 1-2-3 Conjecture claims constructively that every graph with no component isomorphic to $K_2$ admits such a $\{1,2,3\}$ -labelling. A later source attributes a full solution to Keusch, but the retrieved material provides no independent verification or resolved proof audit.

Current status (as of September 2026): A complete proof is claimed by Keusch and in A Solution to the 1-2-3 Conjecture, while the retrieved evidence does not independently verify it.

The 1-2-3 Conjecture for sum-proper edge-labellings

Primary source

Progress summary

Solutions 0