On the Standard (2,2)-Conjecture
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The well-known 1-2-3 Conjecture asserts that the edges of every graph without an isolated edge can be weighted with 1, 2 and 3 so that adjacent vertices receive distinct weighted degrees. This is open in general. We prove that every graph with minimum degree δ≥ 10^6 can be decomposed into two subgraphs requiring just weights 1 and 2 for the same goal. We thus prove the so-called Standard (2,2)-Conjecture for graphs with sufficiently large minimum degree. The result is in particular based on applications of the Lovász Local Lemma and theorems on degree-constrained subgraphs.
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