2019/11/03 by Jakub Przybyło, Przybyło, Jakub
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Graph Labeling and Dimension Problems #Limits and Structures in Graph Theory
paper · doi:10.48550/arxiv.1911.00867
openalex publication_date 2019/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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≥ 106 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.